$G$ 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 nontrivial connected central isogeny onto $G$.

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 $G$.

Welcome for any answer under further assumption that $\text{char }k=0$.