Looks like you have a tension/compression tapping head similar to the Tormach below. I attached the instructions from Tormach in the PDF. They sell a few varieties, instructions for this style are on pages 5-8.
I'm curious how it works out for you. My understanding is the spring "slop" in the head is to allow the spindle to turn at a different rate than the feed advance. Where that's important on the Tormach 770 and 1100 series is the spindle speed is approximate, not 100% accurate. I could set my spindle to turn at 100 RPM and feed the Z at 10 inches per minute and expect to cut 10 threads per inch. But a spindle speed of 100 might be 105 or 95, so I'm not turning the spindle at the same rate as the Z-feed on the head. The T/C head is keyed so the head and arbor turn together, but they can slip on the Z-axis to allow the head to float down or up the arbor. There's around 3/4" float on the Tormach one shown below.
I'm thinking you'll want to leave your tail stock unlocked to the bed in use. Jam the tap in the hole of your chucked work, jog the lathe to advance the tap. Let the tap pull the tail stock along the bed. Stop the spindle and reverse the lathe to back out the tap. A tap held in a drill chuck would work the same, would just take a little longer to change sizes.
Of course, I could be totally off base and maybe what you have is an auto-reverse Procunier type head.
Bruce
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