On Pierce problem for reduced $p$-groups

One of problems in the study of abelian groups and their endomorphism rings is the problem of constructing an appropriate structural theory. The structural theorems obtained for abelian groups and their endomorphism rings allow to reduce the study of complex objects to simpler ones. A radical is one of the tools possessing this property. In this paper we describe the Jacobson radical of the endomorphism rings of reduced $p$-groups from some class, which is a solution to the Pierce problem for this class of groups.