It is a pet peeve of mine that model firms and some authors mix up the Mk 4 and the Mk 12 radars. They use the Mk 12 antenna as the Mk 4 and use the same antenna adding the Mk 22 calling it the Mk 12 radar.
The Mk 4 radar was used on ALL mods of the Mk 37 director (early Taper-back and the Square-back types). The first experimental install was in September 1941 for evaluation and serial production installs started in about November and December 1941. There were faults with the Mk 4 radar at low altitude detection. So a new radar, the Mk 12, operating at a higher frequency was developed. It was also intended to have another height finding radar coupled to it, the Mk 22.
But the early Taper-Back Mod Mk 37 directors (Mod 0 through Mod 7 used on ALL SIMS-BENSON-GLEAVES class destroyers and some cruisers and battleships built just before WWII) could NOT install the new Mk 12 radar due to lack of interior space. So the USN tried installing the Mk 22 radar with the Mk 4 radar on those destroyers with Taper-back Mk 37 directors. It wasn't successful and eventually the Mk 28 radar replaced that combo and any remaining Mk 4 radar equipped units.
On units in the Fleet with Square-Back Mk 37 directors and heavy units like battleships and cruiser with Taper-back Mk 37 directors, which had their director housing modified, the Mk 12 radar started to be installed in Mid-1944. On these directors, box extensions were added for all the "black box" electronics required. The development of the Mk 22 radar was delayed and didn't start to be installed until late in 1944. The Mk 12 radar had a different antenna shape and had a small IFF antenna installed on the antenna face. A very few FLETCHERs operated with only the Mk 12 radar without the Mk 22.
Attached is a good view showing the differences in the shape and size between the Mk 4 radar (left) and Mk 12/22 radar (right) antennas on post-WWII FLETCHER class destroyers awaiting to be mothballed.