# Natural map from suspension of loop induces epimorphism on cohomology?

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 $$G$$?

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 $$r\leq 2n-1$$.