As a concrete example, suppose $Y$ is a connected closed oriented 2-dimensional manifold and $X \to Y$ is a connected cover. Then $H_2(X) = H_2(Y) = H^2(X) = H^2(Y) = \Bbb Z$ by Poincare duality, and the map $H_2(X) \to H_2(Y)$ is multiplication-by-2 because the map is degree 2, so the dual map $H^2(Y) \to H^2(X)$ is multiplication-by-2. The image $2\Bbb Z$ must be contained in the subgroup $H^2(X)^\theta$, which forces $\theta$ to act trivially on $H^2(X)$. However, this means that the image of $H^*(Y)$ does not contain all of $H^*(X)^\theta$.

As abx states in the comments, there is a Hochschild-Serre spectral sequence
$$
H^p(\Bbb Z/2; H^q(X)) \Rightarrow H^{p+q}(Y)
$$
and the map $\pi^*: H^*(Y) \to H^*(X)$ is realized by the "edge morphism" that takes the entries in the column $p=0$. In particular, the image $\pi^*(H^2(Y)) \subset H^2(X)$ is the collection of elements in $H^0(\Bbb Z/2; H^2(X)) = H^2(X)^\theta$ that do not support any differentials in this spectral sequence. However, it is definitely possible to have differentials that land in $H^2(\Bbb Z/2; H^1(X))$ or (a quotient of) $H^3(\Bbb Z/2; H^0(X))$.