Galois group of residue field over $\mathbb Q$

Question

Assume $L$ is a number field and is Galois over $\mathbb Q$,$\mathcal O_L$ is algebraic integer in $L$,$k_L=\mathcal O_L/p\mathcal O_L$ is the residue field,for a prime ideal $(p) \subset \mathbb Z$ inert in $L$,is the natural induced map $Gal(L/\mathbb Q )\to Gal(k_L/\mathbb F_p)$ a isomorphism?I note the orders of the both sides are the same.

Answer 1

Yes, since it is a surjection between finite groups of the same order.