15.64 (2002)

For finite groups $G, X$ define $r(G; X)$ to be the number of inequivalent actions of $G$ on $X$, that is, the number of equivalence classes of homomorphisms $G \to \text{Aut}\,X$, where equivalence is defined by conjugation by an element of $\text{Aut}\,X$. Now define $r_G(n) := \max\{r(G; X) \mid |X| = n\}$.
Is it true that $r_G(n)$ may be bounded as a function of $\lambda(n)$, the total number (counting multiplicities) of prime factors of $n$?