Group property-conditionally retract-potentially characteristic subgroup

Definition
Suppose $$\alpha$$ is a group property, $$G$$ is a group satisfying $$\alpha$$, and $$H$$ is a subgroup of $$G$$. We say that $$H$$ is a strongly image-potentially characteristic subgroup of $$G$$ relative to $$\alpha$$ if there exists a group $$K$$ satisfying $$\alpha$$ and containing $$G$$ as a retract such that $$H$$ is invariant under all the automorphisms of $$G$$ that extend to automorphisms of $$K$$.