What is the inbreeding coefficient for the female offspring of a sib-mating in a haplodiploid system?
From the same article on wikipedia:


  The inbreeding is computed as a percentage of chances for two alleles to be identical by descent. 


Now let's draw a family tree for such mating and sign chances for having each allele:



You can see, that the "brother" has a 50% chance of receiving A allele and the "sister" has a 50% chance of receiving A from their mother, but the "brother" will not have the C allele that the sister certainly has. Their daughter will have the allele from her father and one of mother's alleles. Therefore, her chances of having a C allele are 50%, and for a A or a B allele each 75% (remember, she needs two copies, so we need to end up with 200%; furthermore, she may have two A alleles but not more than one C allele). Chances of her having two A or two B alleles are 56,25% $$0.75 * 0.75 = 0.5625$$

Now we have the percentage of chances for two alleles to be identical by descent, her inbreeding coefficient is 56,25%.

The son will have only one allele, 50% for C, 25% for A and 25% for B. Because he does not have two alleles, by definition there is no inbreeding possible.