In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring $$ H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p) $$ is obtained for any primes $p\geq 2$.

Question: I want to know the cohomology ring $$ H^*(\text{Map}_*(M, S^n);\mathbb{Z}_2) $$ for some manifolds $M$ other than $S^n$, for example, $M=\mathbb{R}^n, \mathbb{R}P^n, \mathbb{C}P^n, \mathbb{T}^n, S^{n-1}\times\mathbb{R}, $ etc. Are there any such generalizations or references? Thanks.