2.78 (1966)

Any set of all subgroups of the same given order of a finite group $G$ that contains at least one non-normal subgroup is called an $IE_{\bar{n}}$-system of $G$. A positive integer $k$ is called a soluble (non-soluble; simple; composite; absolutely simple) group-theoretic number if every finite group having exactly $k$ $IE_{\bar{n}}$-systems is soluble (respectively, if there is at least one non-soluble finite group having $k$ $IE_{\bar{n}}$-systems; if there is at least one simple finite group having $k$ $IE_{\bar{n}}$-systems; if there are no simple finite groups having $k$ $IE_{\bar{n}}$-systems; if there is at least one simple finite group having $k$ $IE_{\bar{n}}$-systems and there are no non-soluble non-simple finite groups having $k$ $IE_{\bar{n}}$-systems).

Are the sets of all soluble and of all absolutely simple group-theoretic numbers finite or infinite? Do there exist composite, but not soluble group-theoretic numbers?