Invariable generation of permutation and linear groups

A subset $\{x_1,x_2,\text{hdots},x_d\}$ of a group $G$ \emph{invariably generates} $G$ if $\{x_1^{g_1},x_2^{g_2},\text{hdots},x_d^{g_d}\}$ generates $G$ for every $d$-tuple $(g_1,g_2\text{hdots},g_d)\in G^d$. We prove that a finite completely reducible linear group of dimension $n$ can be invariably generated by $\lfloor \frac{3n}{2}\rfloor$ elements. We also prove tighter bounds when the field in question has order $2$ or $3$. Finally, we prove that a transitive [respectively primitive] permutation group of degree $n\ge 2$ [resp. $n\ge 3$] can be invariably generated by $O\left(\frac{n}{\sqrt{\log n}}\right)$ [resp. $O\left(\frac{\log n}{\sqrt{\log \log n}}\right)$] elements.