Let $i: C_0 \hookrightarrow C$ be a nil-immersion (a closed immersion defined by a nilpotent ideal sheaf, so it is a homeomorphism of the underlying topological spaces). Denote by $\mathbb{G}_m$ the multiplicative group, and by $\mathbb{G}_{m,C}$ the sheaf on $C$. (Why) is there an isomorphism of étale cohomology groups $H^i(C_0, i^*\mathbb{G}_{m,C}) \cong H^i(C,\mathbb{G}_{m,C})$?
[Milne, Étale Cohomology], p. 30, Theorem I.3.23 might help, which states that there is an equivalence of étale $C$-schemes and étale $C_0$-schemes.