Direct factor is not intersection-closed

This article gives the statement, and possibly proof, of a subgroup property not satisfying a subgroup metaproperty .
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Statement

The intersection of two direct factors of a group need not be a direct factor. In fact, this property is not even true within $p$-groups.

Proof

The counterexample

Let $C_n$ denote the cyclic group on $n$ elements. Then, consider the group $C_p\times C_{p^2}$. Consider the automorphism $\sigma$ of this group which sends the pair $(a,b)$ to $(a+b,b)$. Consider the intersection of the group $C_{p^2}$ (which is a direct factor) with its image under $\sigma$. This is basically the subgroup of $C_{p^2}$ comprising elements which are multiples of $p$. Clearly, this is not a direct factor, in fact it is not even a direct factor inside $C_{p^2}$.