I am not sure if the following is the kind of answer you are expecting, but take the (left) adjoint action $(ad_l h)\cdot k=\sum h_1 kS(h_2)$ of a hopf algebra $H$ on itself. If you set $H=kG$ then the left adjoint action for $g\in G$ reduces to a conjugation: $(ad_l g)\cdot k=gkg^{-1}$.
Now, it is known that the left adjoint action of $H$ on itself is always inner (in the sense of the def. 6.1.1, p. 87, in Montgomery's book "Hopf algebras and their action on rings") and that -since the adj. action of $H$ on itself is inner- then any grouplike element $g\in G(H)$, acts -by conjugation- as an inner automorphism of $H$ (see ch.6, example 6.1.5 in the same book for more details).
Concluding, i would say that the notion of an "inner action" as described in the above reference may be the reasonable notion you are looking for.