Autologism-invariant subgroup for a variety

As an invariance property
A subgroup $$H$$ of a group $$G$$ is termed an autologism-invariant subgroup for a variety $$\mathcal{V}$$ if it is invariant under any bijective set map $$\sigma:G \to G$$ satisfying all three of these conditions:


 * $$\sigma$$ induces an automorphism modulo the marginal subgroup of $$G$$ for $$\mathcal{V}$$, i.e., $$\sigma(xy) = \sigma(x)\sigma(y)$$ mod $$V^*(G)$$, where $$V^*(G)$$ is the marginal subgroup of $$G$$ for $$\mathcal{V}$$.
 * $$\sigma$$ restricts to an automorphism on the verbal subgroup for $$\mathcal{V}$$, i.e., $$\sigma$$ sends the verbal subgroup to itself and the restriction to the verbal subgroup is an automorphism of the derived subgroup.
 * $$\sigma$$ commutes with each of the defining words for $$\mathcal{V}$$.

As a two-case property
A subgroup $$H$$ of a group $$G$$ is termed an autologism-invariant subgroup for a variety $$\mathcal{V}$$ if it satisfies either of these conditions:


 * 1) $$H$$ contains the marginal subgroup $$V^*(G)$$ and $$H/V^*(G)$$ is invariant under any autologism of $$G/V^8(G)$$ that is the marginal factor group part of the data specifying an autologism.
 * 2) $$H$$ is contained in the verbal subgroup $$V(G)$$ and it is invariant under any automorphism of $$V(G)$$ that is the verbal subgroup part of the data specifying an autologism.

Examples

 * For the variety of abelian groups, this becomes the notion of autoclinism-invariant subgroup.