You want to talk about a Spartan empire, the 1st thing you have to get around is traditional Spartan conservatism. Spartans were conservative to a degree that's unimaginable today. Came from being raised in a barracks, conditioned to instant unquestioning obedience to every order. Any change at all from the way great-grampa lived made 'em very nervous. This is why their society hardly changed at all for half a milennium. 500 freakin' years. And they'd never had an empire before, so why would they need one now?
Even when the Spartans conquered Athens, their greatest rival, the issue of colonizing it seems to have never even been raised. What they were divided over was whether or not to destroy it. That would have been for the purpose of keeping things-- in Sparta, at least-- pretty much the way they'd always been.
The Spartans for the most part restricted their military activities to the Pelopennessus. This may have been so they'd be close to home in case of a helot revolt, but maybe not. The evidence is the helots were pretty loyal. Being a Spartan helot, as foreign as this is to our way of thinking, was viewed as a privilege. Even when Xerxes invaded, their solution was for all loyal Greeks to retreat into the Pelopennessus & fortify the Isthmus of Corinth. Become more of those privileged helots, maybe. The other Greeks were unpersuaded.
The size of Spartan society was regulated by birth, & like all Greeks they'd have had a huge problem expanding their definition of citizenship to include anyone whose family hadn't lived in Sparta for 10 generations. This was true even in relatively open-minded Athens, where voting depended on which tribe you belonged to, that also determined which god(s) you worshiped, & both depended on which semi-mythical ancestor you were descended from. If you had none of that, then you just couldn't be an Athenian. Except in Sparta it was worse.
The Spartans did do a little bit of colonizing inside the Pelopennessus. Somewhere I've got a reference that says they colonized a town a few miles over which later provided a 7th mora for their army, so it isn't strictly true they had only 6 morai-- but darned if I can figure out where it is right now or the name of the town.
Thegn.