Function Value Arithmetic Progression

Let α,β and γ be three positive real numbers. Let f(x)=αx5+βx3+γx,x∈R and g:R→R. be such that g(f(x))=x for all x∈R. If a1,a2,a3,...,an be in arithmetic progression with mean zero, then the value of f(g(‌

1

n

‌

n

∑

i=1

f(ai))) is equal to: