at.algebraic topology - Betti Cohomology of singular Kummer Surface

I missed that the question concerned the singular Kummer surface (which I think
historically was what was what was called the Kummer surface but our current fixation on
non-singularity has changed that) so one needs a few more steps than Barth,
Peters, van de Ven: Compact complex surfaces (which will be my reference below).

Let $picolontilde Xrightarrow X$ be the minimal resolution of singularities
and consider the Leray spectral sequence for $pi$. We have $pi_astmathbb
Z=mathbb Z$ and $R^2pi_astmathbb Z$ the skyscraper sheaf with one $mathbb
Z$ at each of the 16 singular points. The Leray s.s. thus gives that
$H^i(X,mathbb Z)=H^i(tilde X,mathbb Z)$ for $ineq2,3$ and hence
$H^i(X,mathbb Z)=mathbb Z$ for $i=0,4$ and $H^1(X,mathbb Z)=0$ as well as a
short exact sequence
$$
0rightarrow H^2(X,mathbb Z)rightarrow H^2(tilde X,mathbb Z)rightarrow
bigoplus_{vin V}mathbb Zvrightarrow H^3(X,mathbb Z)rightarrow0,
$$
where $V$ is the set of singular points. Now, it is easy to see that $H^2(tilde
X,mathbb Z)rightarrow mathbb Zv$ is given by $fmapsto deg(f_{E_v})$, where
$E_v:=pi^{-1}(v)$. We have $deg(f_{E_v})=langle e_v,frangle$, where $e_vin
H^2(tilde X,mathbb Z)$ is the fundamental class of $E_v$. Hence, we get to
begin with that $H^2(X,mathbb Z)$ is the orthogonal complement in $H^2(tilde
X,mathbb Z)$ of the $e_v$. By Cor. 5.6 (of BPV) this can be identified with
$H^2(A,mathbb Z)$. On the other hand, the image of $H^2(tilde X,mathbb Z)$ in
$bigoplus_{vin V}mathbb Zv$ contains the linear functions given by the $e_v$
and $e_v(v')=-2delta_{v,v'}$ so that we may consider the image of $H^2(tilde
X,mathbb Z)$ in $bigoplus_{vin V}mathbb Z/2v$. By the fact that the cup
product pairing on $H^2(tilde
X,mathbb Z)$ is perfect (by Poincaré duality) and by Prop. 5.5 we get that this
image is dual to the subspace of affine functions of $bigoplus_{vin V}mathbb
Z/2v$ (where $V$ is identified by the kernel of multiplication by $2$ in $A$)
and hence we get an identification of $H^3(X,mathbb Z)$ with the dual of the
$mathbb Z/2$-space of affine functions of $V$, in particular it has dimension
$5$.

Remark: It is interesting to note that while the quotient $A/sigma$ as a
topological space does not use the complex structure of $A$ it still seems
easier to use it (in a very weak form, the blowing up only uses that a conical
neighbourhood has a certain form) as we consider the complex blow up of the
singular points. Indeed, the use of Mayer-Vietoris tried by the poser does look
more difficult (of course that would also use the local form of the singularity
but somehow in a less complex fashion).