Let $f$ be a multiplicative arithmetic function. Define $$h(p^2) = f(p)^2 - f(p^2)$$ for any prime $p$. Then $h(p^2) = f^{-1}(p^2)$ for any prime.
Proof. Let $p$ be prime. Then consider $$f * h (p^2) = \sum_{d|p^2} f(d)h(\frac{n}{d}).$$ I think the peoblem does not quite make sence since I don't know how $h(1)$ and $h(p)$ are defined.
Not sure how to proceed from there.