The effect of pressure on the structural behaviour of In alloys with Cd, Sn and Pb has been studied with diamond anvil cells using synchrotron radiation. The face-centred cubic phase (fcc) of an In alloy with 6 at% Cd transforms under pressure into a face-centred tetragonal In-type phase, fct, with c/a>1. The fcc phases of In alloys with 40 and 60 at% Pb transform under pressure to fct with c/a<1. For the ambient pressure fct phase (c/a<1) of an In alloy with 20 at% Sn the distortion increases with pressure. The stability of the tetragonal phases under pressure with respect to the cubic structure is discussed in terms of Brillouin-zone-Fermi-sphere interaction.