People who buy Tikkas are not looking to make a custom rifle out of it. They're looking to get an excellent rifle that shoots stunning groups right out of the box. One reason [a guess] why Empire rifles won't work on Tikka is because the Finnish and Sweedish steel used is so bloody hard that they'd wear out their tools? Just throwing that out there.
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Not at all, they'll work on Swede Mausers, Satterlee's, GMA's, or even colour case hardened actions. All of which are at least as hard and some definitely harder than a Tikka. They won't even thread a barrel for one, as no high quality custom maker I've yet heard of will. The reason is plain and simple, they will not put their name on something with injection moulded plastic parts. Empire Rifles, Martini & Hagn, Satterlee, they all build guns that are good for centuries with proper care, every piece of them from the mag to the bolt to the bottom metal you name it. A Tikka will not last like a Winchester, Ruger, or even Remington will. I risk extending out the argument, but the question was opened to why they won't work on them, and that's why.
I will agree the tikka is a generally quite accurate (with definite exceptions like any brand), serviceable and low priced hunting rifle. I just demand a lot from my guns, far beyond what's required frankly as my son will be using my guns in fifty years, and I have no doubt they'll be just as good as they are today. I wouldn't have that confidence in a tikka. I also find myself going to Africa often now to a conservancy a friend manages, and can't say I trust something like a tikka when I'm trying to determine what the large, close rustle in the bushes is. Or for taking dangerous game for that matter. My Rugers and Winchesters, and my Mauser .275, forgive the piping of those two brands accompanied by Mauser, I do completely.
I no longer own many guns. Well, perhaps in the eyes of some, sure, but I'm down to probably somewhere around ten to twelve keepers from many times that number. I'm working towards just 6 or 7 too, as I'm not rich and rather a thimble of perfect scotch than a bucket of lucky lager. Many with far more means than I would laugh at me calling what I cart around the continents perfection, but to me it is; they shoot perfectly, they take the world's most dangerous game with zero safety concerns, they'll be working the same when my great grand children have them without a doubt, and finally they really didn't come from a different price class than a tikka which would do none of this for certain except likely be accurate.
I often wonder what people are looking for in accuracy in their hunting rifles too. I'm a user of guns, more than a collector, and have averaged more than a half dozen head of game a season these last couple years since my family took root, my career settled, and I've been able to take off into the wilderness more again. I've shot flightless game from 5lbs to 2,000lbs, friendly and dangerous, close and far, and have never needed, nor even noticed the need for, a rifle shooting better than 2" 100 yard groups. My main Ruger hunting rifle handily beats an inch (not an Internet inch), as will my 1946 Winchester Model 70, with the Ruger RSM .375 easily shooting cloverleafs and the Model 70 hanging a little wider at that comfy 1". You know what I found? It makes no difference. I've never kept a hunting rifle I couldn't hit a playing card with at 100 yards, no Ruger 77 or Winchester 70 (or even 94!), even the model 700 I often malign, has ever failed this. I don't punch paper with hunting rifles often, it bores me, I do it enough to practice but it's always from field positions from which we'll never be shooting 0.2 MOA groups. If you find yourself at a bench sitting with a hunting rifle in bags ofts , something's wrong. I enjoy competitive shooting and have engaged in it since I was a kid, back then to the provincial/national level, and super accurate guns have their place- but we're talking hunting guns and even the much compliemnted in accuracy tikka can't stand against true match guns. Yes, there's the tikka tactical that I'm sure has been used to good effect in matches somewhere, but we're pulling out of hunting rifles now.
So, thanks for listening and sorry to go on at length, there's a lot to explain here though. To the OP, pull out the T3's mag, wiggle at its trigger guard, and flick its bolt shroud next time you handle it. For some, myself included, it goes back to the guy behind the counter at that point.