Let $X=S^2\times B^5$ and let us consider an embedding of an open $5$-disk in $X$ with its boundary glued by a map of degree $2$ to some $\{ s_0\} \times S^{4}\subseteq \partial X$ . The embedding fits into a neighborhood of $\{s_0\}\times B^5$ which can be parameterized by $[-1,1]^2\times B^5$ and is defined as follows. We parameterize the first two coordinates of the $5$-disks by polar coordinate system, that is, every point of the $5$-disk is given as a tuple $(r,\alpha,x_1,x_2,x_3)$. This point is embedded to a point $(r(1-r)\sin\alpha, r(1-r) \cos\alpha, r,2\alpha,x_1,x_2,x_3)$. Let $B$ denote the image of this embedding.
The question is as follows: what is the image of the map $H_2\big(X\setminus(\partial X\cup B);\mathbf Z_2\big)\to H_{2}\big(X;\mathbf Z_2\big)$ induced by the inclusion.
