How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?

Question

For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $ I would like to compute the cohomology of $Ri'_*C$. Are there any 'nice' ways to do this? I would to consider the embedding $j: \mathbb{A}^{N}\setminus \{0\} \to \mathbb{A}^{N}$ and restrict $Rj_*C$ to $\mathbb{A}^{N-1}$ instead; yet we do not have a base change isomorphism here, and so the cohomology of $Ri^*Rj_*C$ is not very much related with the one of $Ri'_*C$. Is there any way to 'fix' this nuisance?