Let us work over $\mathbb{C}$. Let $G$ be a finite group, acting on $\mathbb{A}^1$ via a character, and let $H$ be the kernel of the action.

Assume that $\mathbb{A}^1$ is the coarse moduli space of the $[\mathbb{A}^1/G]$, and that $[\mathbb{G}_m/G$ maps onto $\mathbb{G}_m$. The way I picture the stack quotient $[\mathbb{A}^1/G$ is $\mathbb{A}^1$ with a $BG$ at the origin and $BH$ at every point different from the origin. In these terms, how should I think of the difference between $[\mathbb{G}_m/G]$ and $\mathbb{G}_m\times BH$? They ought to be different, but I keep making the mistake of confusing one with the other. Is it possible to write $[\mathbb{G}_m/G]$ as a quotient stack involving some twist of $H$?