Is it true that in general $\operatorname{Gal}(K:\mathbb{Q}_p)\cong\operatorname{Gal}(k:\mathbb{F}_{p^f})$?

Question

I proved that if $K$ is an unramified finite extension of $\mathbb{Q}_p$ of degree $f$, with valuation ring $A$, unique maximal ideal $M$ of $A$ and residue field $A/M\cong\mathbb{F}_{p^f}$, then there is an isomorphism $\operatorname{Gal}(K:\mathbb{Q}_p)\cong\operatorname{Gal}(k:\mathbb{F}_{p^f})$. I used the hypothesis of unramification by using that the extension is generated by the $p^f-1$-th roots of unity. Now I wonder: is the statement still true if we don't suppose that it is unramified?