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 ${\mathcal{M}}_2$ of curves having a degree $n$ elliptic subcover has been computed for $n=3,5$ and discussed in detail for $n$ odd; see [17, 22, 3, 4]. When the degree of the cover is even the case in general has been treated in [16]. In this paper we compute the sublocus of ${\mathcal{M}}_2$ of curves having a degree $4$ elliptic subcover.