Kitchens – Looking for a splashback pattern in kitchen using 3 or 4 different square tile colours

Just in case this helps anyone, I spoke to the people at the tile shop and they recommended building the wall out using cement based tile adhesive (the type you buy as powder and mix). I did this and it worked well. You have to work fast though, that stuff dries very quickly!

A few possibilities...

Probably the cheapest (though not necessarily going to work in your setting) is a 3/4" ship auger drill bit through the plate of the wall behind. A slack jawed yokel would just set the bit at an angle through the drywall and go for it, but a sensible individual would open up a small hole in the drywall beforehand to see what they're drilling through. Hitting plumbing/electrical/etc is not in anyone's best interest.

Next would be a diamond grit hole saw. The only bothersome part of this approach is that you'd have to make a 3/4" hole in a scrap of plywood big enough to stand on. That would guide the hole saw and keep it from walking. If you do this, don't center the 3/4" hole saw exactly on your 1/2" hole. Have the rim of the hole saw go through a little bit of empty space in the 1/2" hole, so that it can drain dust and minimize clogging.

Least appealing would be to use a 3/4" hammer drill bit, but if you have other uses for such a thing (and a hammer drill that can actually drive something that big), it might be an option. On low speed and and not on the hammer setting, make the hole bigger. Too much pressure, and you run the risk of cracking the tile. (Not enough pressure and you'll be there for days.) Once you've opened up the tile, the drill bit won't go through the floor very well, but you can probably push your way through with a little more drill speed. From your description, it doesn't sound like you have porcelain tiles, but if you do, avoid the hammer drill bit like the plague -- I assure you that it won't end well.