D. Naidu constructs an explicit inverse map $H^n(H,A) \rightarrow H^n(G,CoInd_G^H(A))$$H^n(H,A) \rightarrow H^n(G,\operatorname{CoInd}_G^H(A))$ for the case $n=1,2$. For the construction and proof, check out lemma $2.1, 2.2$2.1, 2.2 in his paper https://arxiv.org/pdf/math/0605530.pdfCategorical Morita equivalence for group-theoretical categories.