Equivalence of definitions of fully invariant direct factor

Statement
The following are equivalent for a fact about::direct factor $$H$$ of a group $$G$$:


 * 1) $$H$$ is a fact about::fully invariant subgroup of $$G$$, i.e., every endomorphism of $$G$$ sends $$H$$ to itself. In other words, $$H$$ is a fully invariant direct factor.
 * 2) $$H$$ is a fact about::homomorph-containing subgroup of $$G$$, i.e., for any homomorphism of groups from $$H$$ to $$G$$, the image of $$H$$ is contained in $$H$$.
 * 3) $$H$$ is an fact about::isomorph-containing subgroup of $$G$$, i.e., $$H$$ contains any subgroup of $$G$$ isomorphic to $$H$$.