Section ExtSdprodm.
Variables gT aT rT : finGroupType.
Variables (H : {group gT}) (K : {group aT}) (to : groupAction K H).
Variables (fH : {morphism H >-> rT}) (fK : {morphism K >-> rT}).
Hypothesis actf : {in H & K, morph_act to 'J fH fK}.
Local Notation fsH := (fH \o invm (injm_sdpair1 to)).
Local Notation fsK := (fK \o invm (injm_sdpair2 to)).
Let DgH := sdpair1 to @* H.
Let DgK := sdpair2 to @* K.
Definition xsdprodm := sdprodm (sdprod_sdpair to) xsdprodm_act.
Canonical xsdprod_morphism := [morphism of xsdprodm].

Lemma im_xsdprodm : xsdprodm @* setT = fH @* H * fK @* K.