Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements

For any distinct primes $p$ and $q$, we prove that there is a finite group which does not embed into any finite group invariably generated by an element of order $p$ and an element of order $q$. This gives a negative answer to Problem 21.142 of the Kourovka Notebook \cite{kourovka21}. In fact, for every fixed pair $p,q$, the group $A_n$ cannot embed into such a group once $n$ is sufficiently large in terms of $p$ and $q$.