= Solution
The standard <CW complex> structure on <Real projective space> has one cell in each dimension from zero to three. Its integral cellular boundary is multiplication by $2$ in even positive degrees and zero in odd degrees. Thus the integral <cellular cochain complex> for $\mathbb{RP}^3$ is
$$
\mathbb Z\xrightarrow{\,0\,}\mathbb Z\xrightarrow{\,2\,}\mathbb Z\xrightarrow{\,0\,}\mathbb Z
$$
in degrees $0,1,2,3$. With coefficients $\mathbb F_2$, all its differentials vanish, so every one of these four <cohomology groups> is one-dimensional.
The lift-and-divide construction of the <Bockstein homomorphism> turns the integral differential $2$ into $1$ modulo $2$. Therefore $\beta:H^1\to H^2$ is an isomorphism, while the maps from degrees $0,2,3$ are zero. The <Bockstein cohomology> is consequently
$$
\boxed{H\beta^q(\mathbb{RP}^3;2)=\begin{cases}\mathbb F_2,&q=0,3,\\0,&\text{otherwise}.\end{cases}}
$$
For comparison, in the <mod-two cohomology ring of real projective space> $\mathbb F_2[t]/(t^4)$, $|t|=1$, this says $\beta(t)=t^2$, $\beta(t^2)=0$ and $\beta(t^3)=0$. The last two formulas also follow from the <Bockstein derivation rule> and the truncation $t^4=0$.
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