4WD won't engage. No lights. Just hear relay click

A couple of specific questions:

Does anyone know whether the black cap on the 4WD actuator can be removed to check the actual "magnet part" of the actuator? Or does this potentially cause things to "pop out place" in the transfer box?

Also, I read in one of the documents on here (www.bigjimny.com/mediawiki/index.php/4WD...er_the_car_is_broken) that if no lights illuminate when trying to switch to 4WD then this indicates a problem with the controller. Is this correct, or might it also be the actuator? I think I read in another article that the 4WD light only starts flashing when the 4WD switch on the actuator tells the controller that the box is in 4WD, and then it stops flashing when the hubs are locked successfully.

I can hear the engine note change after the relay click when I press 4WD button, like it is trying to do something.

Sorry for this next stupid question :
Checked the 4x4 bulb in the dash? Does it light up on ignition on, self test?

Yes, the 4x4 and Low ratio lights come on briefly when you start the car, like normal

I think I read in another article that the 4WD light only starts flashing when the 4WD switch on the actuator tells the controller that the box is in 4WD, and then it stops flashing when the hubs are locked successfully.

This is correct, at least on my vehicle. The same when you go into low range, it will flash and beep if it can't complete the selection. Have you tested the connections on the T/B actuator to see if it's recieved the command to engage 4wd? You could also jumper the switch connections on the T/B to fool it into thinking its gone into 4wd and see if it likes that?
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I did try to check that the actuator receives the command to switch to 4WD, but was a bit unsure which wires I need to check.

I already tried jumping the switch that indicates the transfer box is in 4WD. When I do, it then locks the hubs and the lights come up on the dash (not flashing), so it seems my only problem is that the transfer box is not switching across to 4WD

Have a look at this article, there is a wiring diagram
www.forum.suzukiclubuk.co.uk/viewtopic.php?t=13523&start=10

I think the diagram on the left will be for lever and the two on the right are for dashboard switches.

I haven't read the article, but it looks like a similar problem to you.

Robin