Let \(F\) be an algebraically closed field of characteristic \(p>0\) and let \(H\) be a subgroup of a finite group \(G\). Then the set of all elements of the group ring \(FG\) that commute with all elements of \(H\) is called the centralizer algebra \(FG^H\). The purpose of this article is to answer several questions about the algebra \(FG^H\). Among these, the authors provide examples to show that:
1) the centre \(Z(FG^H)\) can be larger that the \(F\)-algebra generated by \(Z(FG)\) and \(Z(FH)\);
2) it is not always true that the simple \(FG^H\)-modules are the same as the nonzero \(FG^H\)-modules \(\operatorname{Hom}_{FH}(S,T\!\downarrow\! H)\), where \(S\) and \(T\) are \(FH\)- and \(FG\)-modules, respectively;
3) \(FG^H\) can have primitive central idempotents that are not of the form \(ef\), where \(e\) and \(f\) are primitive central idempotents of \(FG\) and \(FH\), respectively.