Classes of free group extensions

In this paper we identify different classes of free group extension using core graphs. We show that every free group extension $H\leq K\leq F$ has a base $B$ such that the associated pointed graph morphism $\Gamma_{B}\left(H\right)\to\Gamma_{B}\left(H\right)$ is onto. But if we examine graphs without base points, there is an extension $\left\langle b\right\rangle \leq\left\langle b,aba^{-1}\right\rangle <F_{\left\{ a,b\right\} }$ such that for every base of $F_{\left\{ a,b\right\} }$ the associated graph morphisms are injective.