Power automorphism not implies universal power automorphism

Statement
A power automorphism of a group (i.e., an automorphism that sends every element to a power of itself) need not be a universal power automorphism.

Power automorphism
Let $$G$$ be a group. An automorphism $$\sigma$$ of $$G$$ is termed a power automorphism of $$G$$ if, for every $$g \in G$$, there exists a $$n$$ such that $$\sigma(g) = g^n$$.

Universal power automorphism
Let $$G$$ be a group. An automorphism $$\sigma$$ of $$G$$ is termed a universal power automorphism (or a uniform power automorphism) if there exists a $$n$$ such that $$\sigma(g) = g^n$$ for all $$g \in G$$.

Example of the quaternion group
In the quaternion group:

$$\{ 1,-1,i,-i,j,-j,k,-k \}$$

conjugation by $$i$$ is a power automorphism that is not a universal power automorphism.


 * It is a power automorphism: It fixes $$\{1,-1,i,-i \}$$, and sends the other elements to their inverses.
 * It is not a universal power automorphism: It sends the elements $$\{j,-j,k,-k\}$$ to their inverses. Since each of these elements have order four, any $$n$$ for which this automorphism is the $$n^{th}$$ power map, must be congruent to $$-1$$ mod $$4$$. On the other hand, since it fixes $$i$$, which also has order four, $$n$$ must be $$1$$ modulo $$4$$, a contradiction.