abstract algebra - Generalization of Cauchy's functional equation

We know that if $f(x+y)=f(x)+f(y)$ and $f$ meets some "reasonable" conditions, then $f$ is linear.

I've been considering the following extension: consider the reals under some unknown group operation $\oplus$ which is isomorphic to the reals under standard addition, i.e. $f(x\oplus y)=f(x)+f(y)$. Under what conditions must we conclude that $f(x)=cx$ where $f$ is the isomorphism?

I think over the rationals we could use the same argument as for Cauchy, but I'm not sure about over the reals.

Update: In another question Joy gives an example where $f:(\mathbb{R},\oplus)\to (\mathbb{R},+)$ with $f$ continuous yet $f(x)\not=cx$. So the answer to this generalization is not the same as the answer to the normal Cauchy.