No idea which type they are. No writing or documentation left behind. You may be correct that it was designed to work with Niles. The music has worked fine a few times I have used it before the other day.
This is the wall switch. What are your recommendations now? Do I buy the same Amp as before or is there a better solution to work with this setup?
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Start by disconnecting the amplifier from the SS-6 and turn it on- if it still makes the double click, the amp may have gone to meet its maker. If it clicks once, turn off all of the speakers and connect it, listening for the click- if it clicks differently from when it's not connected, find out why.
Removing the dust and debris in the junction box and on the controls wouldn't be a bad idea and then, show the rest of the control.
If that has the autoformers (they look a bit like small transformers),you should see a pair of small jumpers, made of plastic. They're for connecting more than one pair of speakers and more than one control but if you have a control for each speaker output on the SSS-6, make sure they're appropriate for their location, as in, make sure the one for the outdoor speakers is made for outdoor use, if it's not on an interior wall.
If you're not familiar with how to use a multi-meter, get one and become familiar.
The brand of control doesn't matter- the SS-6 is just a group of switches on a circuit board in a box, with a resistor in series for each channel to make the amplifier happy.