There are several methods. I was a tool and die maker, so this would happen more often for us than other people.
First off, blow compressed air into the hole to clear out any chips and swarf. Blow opposite of the cut direction to force air into the spaces between the tap and base metal.
Aluminum is gummy stuff, so I would WD40 it, and then take a very small punch and a light hammer and tap, tap, tap. Be careful not to mark the surrounding metal. Hold the punch at a very low angle, as close to perpendicular to the tap as possible.
Again, apply compressed air. This pushes the lubricant into the tiny space where the leading edge of the tap is cutting into the base metal. Tap, tap, tap.
Be patient. Keep using these steps until a wiggle is percieved.
If this doesn't dislodge it, use heat. Aluminum expands faster than the High Speed Steel of the tap, so the bond between the tap and base metal will fail. WD40, cool and Tap, tap, tap.....
If it is a 2 flute tap, you can get these things with two prongs into the flutes, and work it that way too.
I once made a tiny "L" shaped tool I could stick in there and use as a a ramrod. It worked pretty well If I do say so myself.
Walk away if you get frustrated. Chill out, tackle later.
i have tried the slow and careful punch route, i also have a narrow pair of long nosed electricians pliers which in some cases can be used to rotate the tap in the hole,though a pronged device can be made easily enough, as already said take it slow and easy, and take a break when you start to get worn by it,
hi how are you getting on? what size is the tap? how deep is it in ? smaller they are harder they are to get out. we break a lot of taps but larger sizes 16 mm plus ,it can be frustrating . the advice given already is good ,proprietry tap extractors are available for 2 3 and 4 flute taps but again patience and persistance is required good luck rick