Let $\alpha , \beta and \gamma$ be three positive real numbers.

$Let f\left(x\right)=\alpha x^5+\beta x^3+\gamma x, x\in R and g : R\to R$be such that $g\left(f\left(x\right)\right)=x for all x\in R.$

$If a_1, a_2, a_3, ..... a_n$ 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: