We consider the subalgebra $Δ$ in the group algebra of the symmetric group $G=S_{n_1+\dots+n_ν}$ consisting of all functions invariant with respect to left and right shifts by elements of the Young subgroup $H:=S_{n_1}\times \dots \times S_{n_ν}$. We discuss structure constants of the algebra $Δ$ and construct an algebra with continuous parameters $n_1$ extrapolating algebras $Δ$, it can be also can be rewritten as an asymptotic algebra as $n_j\to\infty$ (for fixed $ν$). We show that there is a natural map from the Lie algebra of the group of pure braids to $Δ$ (and therefore this Lie algebra acts in spaces of multiplicities of the quasiregular representation of the group $G$ in functions on $G/H$).