For wide body aircraft.PF or PM to push the Auto pilot botton?
Does it really matter?
We do it one of two ways; the pilot flying can request an autopilot, or he can simply reach up and engage it. No difference, really. Our company policy is that no autopilot is engaged until after flap retraction on takeoff, and many of us hand fly until higher altitudes (largely because we don't get much handflying, otherwise). Some will engage it shortly after flap retraction.
Our general policy is that if one is hand flying, one commands the other pilot to do whatever is required, whether it's engaging the autopilot or setting a flight director setting. One may also engage the autopilot. Once the autopilot is engaged, one takes care of the other input parameters, such as the navigation mode (heading, INS, ILS, etc), the vertical mode (altitude hold, airspeed, mach hold, etc), and any autothrottle settings, if applicable. Much of the time we operate with the autothrottle off, and the pilot flying manually works the throttles.
In general, during a takeoff and climb, we don't work the throttles if we're flying. We have a strict company policy regarding autothrottles on takeoff; we don't use them. During the takeoff, if the captain is flying, he removes his hand at the V1 call, and puts both hands on the yoke. Thereafter, he won't manipulate the thrust levers unless he sees a need, during the climb. If the first officer is flying, he sets initial power, then calls for the takeoff thrust ("set reduced thrust," etc), and thereafter doesn't touch the thrust levers during the climb unless he sees a need.
We operate with a flight engineer, and the flight engineer manipulates the thrust levers, setting takeoff thrust and maintaining the thrust during the takeoff, and then setting climb thrust at a thousand feet above field elevation, on departure.
But I feel that PM should do it because PF need to leave his hand from throttles.
I can't imagine that would be the case, but if the pilot flying wants, he can always call for a particular autopilot to be engaged.
We don't have an "autopilot button," per se. We have one of two or three autopilot levers that can be selected to "manual" or "command," and several other mode selectors that are used to tell the autopilot what is desired.
A typical autopilot use for us would be to call for a flight director airspeed mode just after the gear comes up "IAS Hold." If one a heading and then given a direct-to clearance, the pilot not flying will set the FMS, then engage the INS position on the nav mode selector. When a new altitude is assigned, the pilot not flying will set the altitude in the altitude selector, ensure the altitude hold selector is set to altitude select, and then say the new altitude while pointing to it. The pilot flying will also point to the new altitude and repeat it out loud. When the autopilot is desired, the pilot flying may call for it "Autopilot A to Command," or he may simply engage it and announce it "Autopilot A to Command." Not a big deal.
Thereafter, if a heading is assigned, the pilot flying will dial in the new heading, select heading mode, and fly the airplane through the autopilot. If a clearance direct to a new waypoint is assigned, then the pilot not flying will enter it in the nav computer and the pilot flying will confirm it, before the change is executed. The pilot flying will select INS (our nav mode which selects the computers), and off we go. When the flight is approaching within a thousand feet of the cleared altitude, the pilot flying will announce the approach; leaving Thirty Four for Thirty Five," and the pilot not flying will reply "A thousand feet." As the airplane reaches the desired altitude, the pilot flying will select the altitude hold selector from Altitude Select to Altitude Hold.
Likewise, depending on which airplane we're flying, we're confronted with several different types of autohthrottle functions, or no autohthrottle functions. Some of these may be called-for in the climb to alleviate the FE of having to monitor the power, or it may be manually operated by the pilot flying or the FE.
Who actually engages the autopilot...really depends on what the pilot flying wants to do. In the end, it really doesn't matter, so long as it happens when the pilot flying wants it.