Socle is normality-preserving endomorphism-invariant

Statement
The socle of a group (defined as the join of all its minimal normal subgroups) is always a normality-preserving endomorphism-invariant subgroup, i.e., it contains its image under any normality-preserving endomorphism of the whole group. A normality-preserving endomorphism is an endomorphism that sends normal subgroups to normal subgroups.