For $H$ a Hopf algebra, let $V$ be a right $H$-comodule with coaction $\Delta\_R$. Moreover, let $W$ be a subspace of $V$ such that $\Delta\_R(W) \subseteq W \otimes H$, and note that this implies that $\Delta\_R$ restricts to a coaction $V/W \to V/W \otimes H$. If we denote, 
$$
V^H := \lbrace v \in V ~ | ~ \Delta\_R(v) = v \otimes 1 \rbrace,
$$
and analagously
$$
(V/W)^H := \lbrace [v] \in V/W) ~ | ~ \Delta\_R([v]) = [v] \otimes 1 \rbrace,
$$
where $[v]$ denotes the coset of $v$, and $\pi:V \to V/W$ be the canonical projection, then when do we have
$$
\pi(V^H) = (V/W)^H?
$$



Coinvariant Subalgebras of Hopf Comodules and Quotients
Let $\newcommand\Com{\mathsf{Com}^H}\newcommand\Vect{\mathsf{Vect}}\Com$ be the category of right $H$-comodules. This has sufficiently many injectives, so we can compute the right derived functors of the left exact functor $F=\hom\_{\Com}(k,\mathord-):\Com\to\Vect$, where $K$ denotes the trivial, $1$-dimensiona comodule. We can write $\newcommand\Ext{\mathrm{Ext}}\Ext\_{\Com}^p(k,V)=R^pF(V)$. 


Notice that $F(V)=V^H$ for all $V\in\Com$.


If $$\tag{$\star$}0\to W\to V\to U\to0$$ is a short exact sequence in $\Com$, then we have a long exact sequence for the derived functors $R^pF$, which starts with $$0\to W^H\to V^H\to U^H\to\Ext^1\_{\Com}(k,W)\to\cdots$$


We can conclude, then, as usual, that the map $V^H\to U^H$ is surjective if, for example $\Ext^1\_{\Com}(k,W)=0$. This can happen for various reasons: one obvious one is that $W$ be an injective comodule. A draconiant version of this is the condition that $H$ be cosemisimple.


To say something more intelligent, one would probably need to know more details about your concrete situation, though.


Dually, we can use the functor $\newcommand\box{\mathbin{\Box^H}}G=(\mathord-)\box k$, the cotensor product with the trivial module $k$. Here the traditional notation for its derived functors is $\newcommand\Cotor{\mathrm{Cotor}^H}\Cotor\_p(\mathord-,k)=R^pG(\mathord-)$. The long exact sequence for the derived functors of $G$ applied to the short exact sequence $(\star)$ is now
 $$0\to W^H\to V^H\to U^H\to\Cotor\_1(W,k)\to\cdots$$
and we see that for $V^H\to U^H$ to be surjective is enough that $W$ be coflat.


