Old question I know, but it's about evolution and no one pointed out the obvious answer so evolution geek me must speak up.
Short answer is as long as they use haplodiploidy sex determination system they would be expected to want intermittent inbreeding, and this is seen all over the place in insects that use this sex determination system.
Normally the problem with inbreeding is that a parent may be a carrier for a recessive dangerous gene but not have any symptoms of that gene because they have a healthy copy of the gene that is dominate and thus hiding the bad gene. These sort of genes are very common, and there is no easy way to tell if a potential mate has them, or more accurately how many he has since your mate likely is a carrier for many of these sort of recessive genes. Your only saving grace is that each gene is uncommon enough that it's unlikely both parents carry the same recessive gene. but if a father mates with a child who inherited that same bad gene from the father already then they have a 1/4 chance their child will get two copies of the same bad gene, one from the father and one from the mother/daughter, and thus have whatever dangerous condition the 'bad' gene causes. So for every bad gene the father carried there is a 1/8 chance a child with his daughter would end up with whatever disorder the gene caused, once you factor in the odds that the father cares many of these bad genes it becomes a real risk.
However in a haplodiploidy mating system a female can birth a son without mating with a male, the son gets one copy of every gene from it's mother, and no copies from it's non-existent father. This means if the son got any 'bad' gene from his mother he won't have any 'good' copy of that gene from his father to compensate. Any detrimental recessive gene the son gets will cause the son to be born with a defect. That means there is no way a male can be a secret carrier of a bad gene, it should be obviously causing the son harm if he carries it at all.
So as long as the female only mates with healthy sons she can be pretty confident the son isn't a carrier for any nasty recessive genes and so her children can't get two copies of that gene. Thus the biggest risk of inbreeding is pretty much removed by the fact that males can't pass on the bad genes, in fact males help to weed out these sort of genes so even the females are carriers of far fewer then humans are.
There are many kind of aphids which will grow up on a kind of leaf with limited ability to disperse. If two females manage to land on the same leaf they will mate with the sons of the other female, but if no other mates are around a female with birth a bunch of sons then mate with her sons. Notice in this case the females still prefers mating with foreign entities when available, but she will often have to resort to mating with sons and she can do that too. In the aphid's case she is mating with foreign mates when available to get more genetic variety which will make her children more different, and thus more able to adapt to changes in condition in the future.
Now as a different example there are certain breeds of bees where females will preferentially mate with her brother if possible. This is because the brother shares some of the sister's DNA, so the sister's daughters will end up having 75% of her DNA if she mates with her brother, as opposed to their only having 50% of it if she mates with a foreign male. Her kids are less successful at surviving if she mates with her brother, but enough survive that them carrying more of her genetics was a worthwhile tradeoff.
So I explained why these species aren't harmed as much by inbreeding, but why do aphids still prefer foreign mates and bees prefer inbreeding?
Mostly this comes down to a trade off between lack of genetic variety and advantage of sharing more DNA with your children. Too little genetic variety from foreign mates will lead to your children not being adaptable and thus all of then dying out if the environment changes. In fact lots of clonal species have evolved only to go extinct again for this very reason, the clonal mothers were able to pass on twice their genetics to their children which helped in the short term, in the long term lack of gene flow to support adaptation almost always kills these species pretty 'quickly', at least relative to an evolutionary time scale, as soon as something changes in their home environment.
This means even without the harm of recessive genes an insect would want to mate with foreign mates sometimes to get genetic variety. However, an insect also benefits from her children having more of her DNA that comes from inbreeding. The parent therefore needs to keep a careful balancing act, trading off inbreeding with outbreeding to get the right trade off between having enough foreign genes for her children to survive while keeping as much of her family's genetics to ensure the children pass on as much of her gene as possible. The more frequent the inbreeding the more important it is to outbreed, and vice versa.
In the case of the aphids they often have to inbreed when they have no mates, so a certain percentage of inbreeding is guaranteed in their lineage no matter what, meaning when an opportunity to mate with a foreign individual is available they take it because they likely are in need of more outbreeding.
For the bees the female prefers mating with her brother to an extent, but only so far. A stronger foreign male may be preferable to a weak brother. The degree of preference to mating with her brother is part of the same balancing act. It ensures that females end up choosing to mate with their brothers at roughly the correct frequency to keep the perfect mixture of inbreeding and outbreeding.
So in your species rather or not they engage in inbreeding would likely come down to how much inbreeding is in their immediate ancestor. If your father is also your grandfather and your great uncle it's probably time to find a foreign mate. If your family tree is starting to actually look like a tree then you probably want to get in on some of that sweet sweet inbreeding action.