Kourovka Notebook Problem 21.142

A group $G$ is said to be invariably generated by $a$ and $b$ if $G$ is generated by the conjugates $a^g,b^h$ for every $g,h$. Let $p\ne q$ be fixed primes. Does every finite group embed into a finite group invariably generated by an element of order $p$ and an element of order $q$?