Given an action of a finite group $\Gamma$ on a polynomial ring $k[X_1,\ldots, X_n]$, we define the Reynolds operator $R_\Gamma: {k}[X_1,\ldots, X_n]\to {k}[X_1,\ldots, X_n]$ by $$ R_\Gamma(f)=\frac{1}{\lvert \Gamma\rvert}\sum_{\gamma\in \Gamma}\gamma\cdot f.$$ This operator satisfies some nice properties, in that it is ${k}-$linear, and it projects ${k}[X_1,\ldots, X_n]$ onto its ring of invariants under $\Gamma-$action: ${k}[X_1,\ldots, X_n]^\Gamma.$
My question is the following: how is $R_\Gamma$ defined for when ${k}\ne \mathbf{Q,R,C}$? That is, we could conceivably run into a scenario over $\mathbf{F}_p$ (or some other general field) where $\frac{1}{\lvert \Gamma\rvert}$ does not make any sense, and is not cancelled by a term of $\lvert \Gamma\rvert$ popping out of the sum.