Let A and B be modules, which are faithfully flat over their endomorphism ring. The categories of A-solvable and B-solvable modules coincide if and only if A and B are similar. While similar modules have Morita equivalent endomorphism rings, the failure of the converse raises the question which module-theoretic properties are shared by modules with equivalent endomorphism rings. This paper addresses this question by investigating equivalences between full subcategories of the categories of A-and B-solvable modules, respectively. In particular, every equivalence between the category of A-solvable and the category of B-solvable modules is induced by a Morita equivalence between EA and EB if A and B are faithfully flat as modules over their endomorphism ring. Several examples show that these results may fail without the faithfulness condition.