  Let $M$ be a compact oriented $d$-dimensional smooth manifold and $X$ a
topological space. Chas and Sullivan \cite{Chas-Sullivan:stringtop} have
defined a structure of Batalin-Vilkovisky algebra on
$\mathbb{H}_*(LM):=H_{*+d}(LM)$. Getzler \cite{Getzler:BVAlg} has defined a
structure of Batalin-Vilkovisky algebra on the homology of the pointed double
loop space of $X$, $H_*(\Omega^2 X)$. Let $G$ be a topological monoid with a
homotopy inverse. Suppose that $G$ acts on $M$. We define a structure of
Batalin-Vilkovisky algebra on $H_*(\Omega^2BG)\otimes\mathbb{H}_*(M)$ extending
the Batalin-Vilkovisky algebra of Getzler on $H_*(\Omega^2BG)$. We prove that
the morphism of graded algebras
$$H_*(\Omega^2BG)\otimes\mathbb{H}_*(M)\to\mathbb{H}_*(LM)$$ defined by Felix
and Thomas \cite{Felix-Thomas:monsefls}, is in fact a morphism of
Batalin-Vilkovisky algebras. In particular, if $G=M$ is a connected compact Lie
group, we compute the Batalin-Vilkovisky algebra $\mathbb{H}_*(LG;\mathbb{Q})$.
