What would be blood groups of the progeny of father A and mother homozygous B

The correct option is D AB and B
The blood group of the progeny will be AB if the father has a genotype I${}^A$I${}^A$. In such a case the allele I${}^B$ is inherited from the mother and I${}^A$ from the father and the resulting blood group is AB. Or it can be B if the father has a genotype of I${}^A$I${}^O$. In such a case if the allele I${}^O$ is inherited from the father and I${}^B$ from the mother then the resulting progeny will have blood group B with genotype I${}^B$I${}^O$. So, the correct option is 'AB and B'.