Kernel systems on finite groups

We introduce a notion of kernel systems on finite groups: roughly speaking, a kernel system on the finite group G consists in the data of a pseudo-Fro benius kernel in each maximal solvable subgroup of G, subject to certain na tural conditions. In particular, each finite CA-group can be equipped with a canonical kernel system. We succeed in determining all finite groups with kernel system that also possess a Hall p'-subgroup for some prime factor p of their, order; this generalizes a previous result of ours (Communication s in Algebra 18(3), 1990, pp. 833-838). Remarkable is the fact that we make no a priori abelianness hypothesis on the Sylow subgroups.