For \(S\) a completely regular semigroup, let \(C^*(S)\) be the least subsemigroup of \(S\) that contains the set \(E(S)\) of idempotents of \(S\) and contains \(a^{-1}ta\) whenever it contains \(t\) (where \(a^{-1}\) is the \(\mathcal H\)-related inverse of \(a\)). (The author shows that this is equal to the `self-conjugate core' of \(S\), in the sense of \textit{P. G. Trotter} [J. Algebra 137, No. 1, 166-179 (1991; Zbl 0714.20053)].) It is shown that \(C^*(S)\) is the analogue of the `Type-II kernel' of finite semigroup theory; that is, it is the intersection of the inverse images of the identity element, over all `completely regular' relational morphisms to groups. (In a note added in the proof, it is acknowledged that this result can be derived from work of \textit{D. B. McAlister} [J. Aust. Math. Soc., Ser. A 29, 475-503 (1980; Zbl 0439.20038)].) In varietal terms, this theorem is equivalent to the equation \(\langle{\mathcal U} \circ {\mathcal G}\rangle = {\mathcal U}C^*\), where \(\mathcal U\) is any variety of completely regular semigroups and \(\mathcal G\) is the variety of groups. Here \({\mathcal U} \circ {\mathcal G}\) is the Malcev product of \(\mathcal U\) with \(\mathcal G\), \(\langle\) \(\rangle\) denotes generation of varieties and \({\mathcal U}C^*\) comprises the completely regular semigroups \(S\) for which \(C^*(S) \in {\mathcal U}\). Some consequences for associativity of Malcev products are derived. \{Reviewer's remark: Recently, \textit{P. G. Trotter} [``Covers for regular semigroups and an application to complexity'' (to appear)] has proved a similar theorem for arbitrary regular semigroups\}.