Other ways to make a square socket, in addition to the good ways already mentioned:
1. drill a hole in a round steel bar that fits over the corners of the square stock. Heat to bright red. Swage to a socket shape with hammer over the end of the 3/16 square bar. The square bar will likely get damaged as you go, just cut off the damaged bit and continue.
2. Google "rotary broach" and make one, use that.
3. drill a 3/16" round hole in a round steel bar. Find a piece of 3/16" square high speed steel. Sharpen the high speed steel end to a sort of reverse pyramid or cone shape to make a mortising punch. hammer the punch into the hole to shave out the corners. (some hex socket wrenches are made this way). Some sort of sleeve for guidance from punch to workpiece will help.
4. Build an EDM machine and use that to burn the square socket. (this sounds far fetched but it really is not difficult if you have built any electronics projects. Google "how to build an EDM" , a book by Ben Fleming. I built his original design machine and it works really well for any sort of odd shaped hole, and for removing broken drills, bolts, taps etc from holes in anything.)
Just food for thought. As usual, there are lots of ways.