Summation of a Modular Formula

For an odd prime $p$, define $f(p)= \lfloor 2^{2^p} \rfloor \bmod 2^p$
For example, when $p=3$, $\lfloor 28/3\rfloor =85\equiv 5(\bmod8)$ and so $f(3)=5$.

Further define $g(p)=f(p)\bmod p$. You are given $g(31)=17$.

Now define $G(N)$ to be the summation of $g(p)$ for all odd primes less than $N$.
You are given $G(100)=474$ and $G(10^4)=2819236$.

Find $G(10^7)$