Property of Group Automorphism which Fixes Identity Only/Corollary 3

Corollary to Property of Group Automorphism which Fixes Identity Only
Let $G$ be a finite group whose identity is $e$.

Let $\phi: G \to G$ be a group automorphism.

Let $\phi$ have the property that:
 * $\forall g \in G \setminus \set e: \map \phi t \ne t$

That is, the only fixed element of $\phi$ is $e$.

Let:
 * $\phi^2 = I_G$

where $I_G$ denotes the identity mapping on $G$.

Then $G$ is an abelian group of odd order.

Proof
Let $s, t \in G$.

Then:

$G$ is of even order.

Then from Even Order Group has Order 2 Element:

But this contradicts the condition on $\phi$:
 * $\forall g \in G \setminus \set e: \map \phi t \ne t$

Hence there is no such element of $G$ of order $2$.

Thus $G$ must be of odd order.