Genus two curves covering elliptic curves have been the object of study of many articles. For a fixed degree $n$ the subloci of the moduli space $\M_2$ of curves having a degree $n$ elliptic subcover has been computed for $n=3, 5$ and discussed in detail for $n$ odd; see \cite{Sh1, SV2, Fr, FK}.

When the degree of the cover is even the case in general has been treated in \cite{PRS}. In this paper we compute the the sublocus of $\M_2$ of curves having a degree $4$ elliptic subcover.