For an odd prime $p$, define $f(p)=\lfloor \frac{2^{(2^p)}}{p}\rfloor mod{2}^p$
For example, when $p=3$, $\lfloor 2^8/3\rfloor =85\equiv 5(\mathrm{mod}8)$ and so $f(3)=5$.

Further define $g(p)=f(p)modp$. 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 $G({10}^7)$.