Is PRG On Concatenation of Input a PRG
I think I figured out the solution.

Assume such G exists (a PRG).

Then G' as described here is a PRG as well.

But applying it on $x\|x$ results in $G'':x\mapsto G\left(x\right)\circ G\left(x\right)$, which is not a PRG (easy to construct a distinguisher that checks if its input $z=z'\|z'$).

It is also possible to use $G′:\left(x\|y\right)↦x\|G\left(y\right)$ instead.