Let F be an algebraically closed field, G be a finite group and H be a subgroup of G. We answer several questions about the centralizer algebra FG(H). Among these, we provide examples to show that
the centre Z(FG(H)) can be larger than the F-algebra generated by Z(FG) and Z(FH),
FG(H) can have primitive central idempotents that are not of the form e f, where e and f are primitive central idempotents of FG and FH respectively,
it is not always true that the simple FG(H)-modules are the same as the non-zero FG(H)-modules Hom(FH)(S,T down arrow H), where S and T are simple FH and FG-modules, respectively.