If $\varphi : A\to B$ is surjective, then is $GL_{0}\left(B\right)$ contained in $\varphi\left(GL\left(A\right)\right)$?

Question

Let $A$ and $B$ be two $C^*$-algebras. Suppose $\varphi : A\to B$ is a surjective $*$-homomorphism. Denote the set of invertible elements of $A$ by $GL\left(A\right)$ and denote the set of invertible elements ,whict are in the same connecting part with $1$, of $B$ by $GL_{0}\left(B\right)$. Then $GL_{0}\left(B\right)$ is contianed in $\varphi\left(GL\left(A\right)\right)$?