The theory $\mathrm{ACA}_0$ is not reflexive (because it is finitely axiomatisable and cannot prove its own consistency). So how, if at all, is it possible to prove that $\mathrm{Q+Con(PA)}$ cannot be interpreted in $\mathrm{ACA}_0$?

(I am assuming that for an interpretation of the second-order language of arithmetic, the domain of the number variables must be interpreted to be a set of numbers which is definable without parameters.)

The theory $\mathrm{ACA}_0$ is not reflexive (because it is finitely axiomatisable and cannot prove its own consistency). So how, if at all, is it possible to prove that $\mathrm{Q+Con(PA)}$ cannot be interpreted in $\mathrm{ACA}_0$?

(I am assuming that for an interpretation of the second-order language of arithmetic, the domain of the number variables must be interpreted to be a set of numbers which is definable without parameters.)

Edit: I believe this is the answer. Because $\mathrm{ACA}_0$ can prove the $\Sigma^{0}_{1}$-completeness of $\mathrm{Q}$. So if the relativisation of $\mathrm{Q+Con(PA)}$ to some definable class model of $\mathrm{Q}$ were provable in $\mathrm{ACA}_0$, then $\mathrm{Con(PA)}$ would also be provable in $\mathrm{ACA}_0$, but this is impossible by Gödel's second incompleteness theorem. Is this correct?