Summation of a Modular Formula

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Problem 717

For an odd prime $p$, define $f(p) = \left\lfloor\frac{2^{(2^p)}}{p}\right\rfloor\bmod{2^p}$
For example, when $p=3$, $\lfloor 2^8/3\rfloor = 85 \equiv 5 \pmod 8$ 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 $G(10^7)$