  For a non-empty class of groups $\cal L$, a finite group $G = AB$ is said to
be an $\cal L$-connected product of the subgroups $A$ and $B$ if $\langle a,
b\rangle \in \cal L$ for all $a \in A$ and $b \in B$. In a previous paper, we
prove that for such a product, when $\cal L = \cal S$ is the class of finite
soluble groups, then $[A,B]$ is soluble. This generalizes the theorem of
Thompson which states the solubility of finite groups whose two-generated
subgroups are soluble. In the present paper our result is applied to extend to
finite groups previous research in the soluble universe. In particular, we
characterize connected products for relevant classes of groups; among others
the class of metanilpotent groups and the class of groups with nilpotent
derived subgroup. Also we give local descriptions of relevant subgroups of
finite groups.
