For the sake of completeness, let's look at the following simple example of very nice continua, and of their inverse system of inclusions, for which there is no homological surjection.
All continua are subspaces of $\ \mathbb R^2$. Let $\ X:=(0\ 0)\ $ be a single-point space. Let $\ C(p;r)\ $ and $\ B(p;r)\ $ stand for closed and open discs which have their center in $\ p\ $ and their radius equal to $\ r.\ $ Define:
$$X_n\ \,:=\ \,C\left(\left(2^{-n}\ 0\right);\ 2^{-n}\right)\ \setminus\ B\left(\left(\frac 3{2^{n+1}}\ 0\right);\ 2^{n+1}\right)$$
Thus $\ H_*(X)=0\ $ while $\ H_*(X_n) = H_*(S^1).\ $ As we see, there are no surjections.(homological)