Stability group of subnormal series of finite group has no other prime factors

This article states and (possibly) proves a fact about a finite group and a Coprime automorphism group (?): a subgroup of the automorphism group whose order is relatively prime to the order of the group itself.
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Statement

Let $G$ be a finite group, and consider a subnormal series of $G$: $\{ e \} \triangleleft H_1 \triangleleft H_2 \triangleleft \dots \triangleleft H_{n-1} \triangleleft H_n = G$

Then, any prime factor of the order of the stability group of this subnormal series, must divide the order of $G$. In other words, no non-identity stability automorphism of this subnormal series can have order relatively prime to the order of $G$.

Related facts

Converse

The converse to the statement is not true: we can have a subgroup of the automorphism group of $G$, with no other prime factors to its order, which cannot be realized as the stability group of any subnormal series. However, the converse is true if we restrict ourselves to $G$ a group of prime power order:

Facts used

1. Centralizer of coprime automorphism in homomorphic image equals image of centralizer