Let $G/S$ be a group scheme (i'm fine if you want to assume everything is affine).
Let $$g : S \to G \in G(S)$$ be an $S$ point of $G$. We can define a morphism $$\phi_g : G \to G$$ defined at the level of points by $x \mapsto g$
Let $\omega \in \Gamma(G,\Omega^1_{G/S})$ be a global differential form on $G$, is it true that $\phi_g^* \omega = 0$ ?
An example is $G = \mathbf{G}_{a, S}$ over $S = \text{Spec}(A)$ and $g$ is given by $a \in A$. Then $\phi_g(x) = a$, i.e., $\phi_g$ is given by the $A$-algebra map $A[x] \to A[x]$ which sends $x$ to $a$. Then $\text{d}x$ maps to zero (in relative differentials) hence it is true in this case.
It is true in general too, because $\phi_g$ factors as $G \to S \to G$ and hence your map $\phi_g^* : \Gamma(G, \Omega_{G/S}) \to \Gamma(G, \Omega_{G/S})$ factors through $\Gamma(S, \Omega_{S/S})$ which is zero.