Question on the Definition of Automorphism

So I am missing something very basic here, but here it goes. As a consequence of the definition of a homomorphism we must have that $f(e) = e’$. Consider $(mathbb{Z}/5mathbb{Z})^*$ and the map $x to 2x$. It is a bijection but does not satisfies the multiplicative property but how can we claim it is an automorphism since $f(1) = 2 neq 1$.

Edit: Does not satisfy multiplicative property.