Are isomorphisms automorphisms? Suppose we have a set $X$ and two group operations on the set $\oplus,+$. If they are isomorphic, does it follow that $\oplus=+$? I am unable to think of counter-examples, and this makes sense to me at an intuitive level because "isomorphism" means something in my head like "the operations are the same, just with a different set." However, I'm unable to prove it using the standard definition of $f(x\oplus y)=f(x)+f(y)$.

No. Let $G$ be a nontrivial group under the operation $\bullet$ and let $f:G\to G$ be a permutation of its elements (which we don't regard as a homomorphism). Define $\circ$ on $G$ by $f(a\circ b)=f(a)\bullet f(b)$.

Then $(G,\circ)\cong(G,\bullet)$ but $\circ,\bullet$ needn't be the same operation. Essentially this means we can make the underlying set of $G$ into the group $G$ in many different ways by relabelling its elements.