$G$ is a semisimple algebraic group over $k$, if $G_{\bar k}$ is simply connected when we do base change to $\bar k$, can we descent the simply connectedness to $G$?

Here, simply connectedness means no connected etale algebraic group cover.

Can we say that simply connected algebraic group is geometrically connected? If then we can give an affirmative answer by considering the universal cover of $G$.