Equivalence of definitions of variety-containing subgroup of finite group

Statement
The following are equivalent for a subgroup $$H$$ of a finite group $$G$$:


 * 1) The subgroup is a fact about::variety-containing subgroup: Let $$\mathcal{V}$$ be the subvariety of the variety of groups generated by the group $$H$$. Then, for any subgroup $$K$$ of $$G$$ in $$\mathcal{V}$$, $$K$$ is contained in $$H$$.
 * 2) The subgroup is a fact about::subhomomorph-containing subgroup: $$H$$ contains any subgroup of $$G$$ that occurs as a homomorphic image of a subgroup of $$H$$.
 * 3) The subgroup is a fact about::subisomorph-containing subgroup: $$H$$ contains any subgroup of $$G$$ that is isomorphic to a subgroup of $$H$$.

Related facts

 * Equivalence of definitions of variety-containing subgroup of periodic group: The proof can be generalized to any periodic group, i.e., a group where every element has finite order.
 * omega subgroups not are variety-containing
 * Variety-containing implies omega subgroup in group of prime power order

Opposite facts

 * Subisomorph-containing not implies subhomomorph-containing
 * Subhomomorph-containing not implies variety-containing