Almost all wreath product character values are divisible by given primes

For a finite group $G$ with integer-valued character table and a prime $p$, we show that almost every entry in the character table of $G\wr S_N$ is divisible by $p$ as $N\to \mathrm{\infty }$. This result generalizes the work of Peluse and Soundararajan on the character table of $S_N$.