Mxsmanic wrote:
Can you use an autopilot when training for or being examined for an
instrument rating, or do you have to fly the aircraft by hand?

There are varieties of "autopilot". 1,2,3-axis, coupled to GPS,
coupled to ILS, wing-leveler, etc.

Me, personally, have a 1/2-axis autopilot. Turn it on, it holds
the heading. Well, not really. Since it's not coupled to anything,
it doesn't know when there's any deviation, such as winds. I can
start out on a heading of 120, turn on the AP. If winds
are from the north, then my actual track is more southerly, depending
on the strength of the winds.

As for IFR training -- you betcha! It's one way to reduce the load on
the student. Need to get charts for an unexpected diversion? Turn on
the AP, reach over and get the charts and start looking for the
right one.

Now, the DE can fail the AP during the exam. In exams, almost anything
is fair game.