The image of an ideal under an automorphism is a subset of the same ideal

Question

Let $A$ be a commutative ring with an identity element $1$, let $a$ be an ideal of $A$, and $f: A \rightarrow A$ be any automorphism. Is it true that $ f(a)\subseteq a$.

Answer 1

Another (counter)example: take $A = \mathbb Z[i]$, $a = (2+i)A$, and $f$ to be complex conjugation.

Answer 2

A sort of universal counterexample is $A=\mathbb Z[X,Y]$, $f$ the automorphism exchanging $X$ and $Y$ and $\mathfrak a=(X)$.