On Fomenko's Homotopical topology page 335, we have the map $\pi(X):\Sigma\Omega X\to X$ induces an epimorphism $\pi(X)_*: H_r(\Sigma\Omega X)\to H_r(X)$. I wonder whether this is true for cohomology too? Specifically, is it true that $\pi(X)^*: H^r(X, G)\to H^r(\Sigma\Omega X, G)$ is surjective for any $G$?
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No.
If $X=S^1$, then $\Omega S^1$ is homotopy equivalent to a countable discrete space, while the suspension of a countable discrete space is a countable wedge of circles. Thus
$$\Sigma\Omega S^1\simeq\bigvee_\mathbb{N}S^1$$
The space $S^1$ is $0$-connected, but $H^1(S^1)\rightarrow H^1(\bigvee_\mathbb{N} S^1)$ is not surjective, since the target group is infinitely generated.
On the other hand, for any $(n-1)$-connected CW complex $X$ we can use the Universal Coefficient Theorem to see that $H^r(X;G)\rightarrow H^r(\Sigma \Omega X;G)$ is injective for any $r\leq 2n-1$.
