This seems obvious in view of the relation of $\mathsf{PA}$ and $\mathsf{ACA}_0$, but I just want to make sure that I'm not overlooking something important.
Edit: This is false in view of the fact that $\mathsf{ACA}_0$ can be reduced to $\Sigma_1^0$-$\mathsf{CA}_0$ with second-order parameters (see: https://www.logicmatters.net/2007/09/14/aca0-3-finite-axiomatizability/, also Simpson, Subsystems of SOA, III.1.3).
New question: What about if there are no free set variables allowed in the comprehension scheme?
The background of this question is that I'm trying to understand what more one (implicitly) assumes when one moves from $\mathsf{I}\Sigma_n^0$ to $\mathsf{I}\Sigma_{n+1}^0$. My conjecture was that it has something to do with the sets of natural numbers that one implicitly assumes to exist...