Should I use oil primer underneath oil paint

Primers (of either base) can be top coated with either latex or oil. Double check your primer label.

What doesn't work is an oil based topcoat (brittle film) over a latex topcoat (softer, flexible film).

An oil based primer is a good undercoat for exterior (or baths), due to its greater degree of waterproofing.

Alkyd over latex is touch-and-go - well, it works or it doesn't. That depends on several things.

First, is the latex paint new? If it's less than a year old, forget it - it will not be happy being overcoated with an alkyd (oil based) paint.

Second, you can do an adhesion test. If it passes the adhesion test, you are all set - you can use that paint. Google's not much help here because everybody wants to sell you elaborate paint-test equipment. You only need a razor blade, packing tape, and a cotton ball. And this procedure explains it. The test is fairly over-wrought, but you can shortcut down to what matters to you. I've bolded the parts that I'd do.

• Find an area which is concealed and won't show too badly, but is intact.
• Make a cross-hatch of razor cuts through the first few coats of paint.
• Clean it carefully. Don't leave any soap residue.
• Do a tape test: stick tape firmly to the crosshatch area, and pull it off, folding the tape back nearly 180 as you pull it off (not straight out). These first few steps are documented in this video. Also it's better to use packing tape.
• Tape a cotton-ball to the cross-hatched area, and wet it in the paint's reducer (i.e. paint thinner). Use very long runs of tape, as the thinner may try to dissolve the tape goo.
• Wait 30 minutes, remove it and see if the paint has softened or dissolved.
• Wait til it dries.
• Do the tape yank test again.
• Sand, wipedown and prep the crosshatch area same as you plan to prep the house.
• Paint it with that paint.
• Wait til it dries. Look for problems.
• Do the tape yank test again.