Let $\alpha ,\beta$ and $\gamma$ be three positive real numbers. Let $f\left(x\right)=\alpha {\mathrm{x}}^5+\beta {\mathrm{x}}^3+\gamma \mathrm{x},\mathrm{x}\in \mathrm{R}$ and $g:R\to R$ be such that $g\left(f\left(x\right)\right)=x$ for all $x\in R$. If $a_1,a_2,a_3,\dots ,a_n$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\frac{1}{n}\sum _{i=1}^{n}f\left(a_i\right)\right)\right)$ is equal to