When lead storage battery is charged?

During charging of a lead storage battery, the reaction at the anode and cathode are
$$\begin{gathered} {\mathrm{Anode}: \mathrm{PbSO}}_4+2{\mathrm{e}}^{-}⟶\mathrm{Pb}+{\mathrm{SO}}_4^{2-} \\ \text{ Cathode: }{\mathrm{PbSO}}_4+2{\mathrm{H}}_2\mathrm{O}⟶{\mathrm{PbO}}_2+4{\mathrm{H}}^{+}+{\mathrm{SO}}_4^{2-}+2{\mathrm{e}}^{-} \end{gathered}$$
In both the reactions H₂SO₄ is regenerated.