Splitting automorphism of prime order

Definition
Let $$p$$ be a prime number.

An automorphism $$\phi$$ of a group $$G$$ is termed a splitting automorphism of prime order $$p$$ if:

$$\phi^p = 1$$

and:

$$x \phi(x) \phi^2(x) \ldots \phi^{p-1}(x) = e$$

for all $$x \in G$$. Note that the definition does not exclude the case of the identity map.