jmc

If an integer $n$ has three digits in base $5$, then $5^2\le n<5^3$. If an integer $n$ has two digits in base $8$, then $8^1\le n<8^2$. The overlap of these intervals is $$\{25,26,27,28,\dots ,61,62,63\}.$$The average of the integers in this set is $\frac{25+63}{2}=\boxed{44}$.