Solvable radical is normal-homomorph-containing

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., solvable radical) always satisfies a particular subgroup property (i.e., normal-homomorph-containing subgroup)}
View subgroup property satisfactions for subgroup-defining functions $|$ View subgroup property dissatisfactions for subgroup-defining functions

Statement

Suppose $H$ is the solvable radical of a group $G$, i.e., the unique largest solvable normal subgroup of $G$. (Note: The solvable radical may not exist for every group. It does, however, exist for finite groups, virtually solvable groups, and slender groups).

Then, $H$ is a normal-homomorph-containing subgroup of $G$. In other words, for any homomorphism $f:H\to G$ such that $f(H)$ is a normal subgroup of $G$, $f(H)\leq H$.

Related facts

Stronger facts: stronger subgroup properties satisfied

Property	Meaning	Proof of satisfaction by solvable radical	Why it's stronger than normal-homomorph-containing
Normal-subhomomorph-containing subgroup	contains any normal subgroup that is a homomorphic image of a subgroup	solvable radical is normal-subhomomorph-containing	obvious

Weaker facts: weaker subgroup properties satisfied

Property	Meaning	Proof of satisfaction by solvable radical	Why it's weaker than normal-homomorph-containing
weakly normal-homomorph-containing subgroup	invariant under homomorphisms from it to whole group that send normal subgroups to normal subgroups	normal-homomorph-containing implies homomorph-containing
Strictly characteristic subgroup	invariant under all surjective endomorphisms	solvable radical is strictly characteristic	normal-homomorph-containing implies strictly characteristic
Characteristic subgroup	invariant under all automorphisms	solvable radical is characteristic	(via strictly characteristic)
Normal subgroup	invariant under all inner automorphisms	solvable radical is normal	(via characteristic)