Let $Y$ be an oriented manifold of dimension three, and let $X=Y\times S^1$. We have

$$H^2(X,\mathbb{Z}_2)=H^2(Y,\mathbb{Z}_2)\oplus H^1(Y,\mathbb{Z}_2).$$

Pick an element $m\oplus n\in H^2(Y,\mathbb{Z}_2)\oplus H^1(Y,\mathbb{Z}_2)$, and consider $\mathfrak{P}(m\oplus n)$, where $\mathfrak{P}$ is the Pontryagin square operation. When evaluated on the fundamental class $[X]$, it is an integer modulo 4, and I believe it is even

$$[X]\frown\mathfrak{P}(m\oplus n) =2k_Y(m,n)$$

where $k_Y(m,n)$ is defined modulo 2. I'd like to know more concrete explicit expression of $k_Y$.

(I think that it's of the form

$$k_Y(m,n)=([Y]\frown(m\smile n)) + \mathfrak{Q}_Y(n)$$

where $\mathfrak{Q}(n)$ is a quadratic form on $H^1(Y,\mathbb{Z}_2)$. If it's true, is there a more explicit expression for $\mathfrak{Q}_Y(n)$? The only thing I can think of is $[Y]\frown(n\smile\beta(n))$, where $\beta$ is the Bockstein. )