According to Serebii.net, it learns Extremespeed at level 39 if it's in Diamond/Pearl/Platinum or HeartGold/SoulSilver.
However in Ruby/Sapphire/Emerald, it learns it at level 49.
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Growlithe does not learn Extremespeed, only its evolved form, Arcanine, learns Extremespeed at level 49. Note that this is the same level that Growlithe learns flamethrower, which Arcanine can only learn through a TM, so if you want both Extremespeed and Flamethrower, evolve Growlithe at level 49. Source: serebii.net
Arcanine doesn't learn flamethrower in any Pokemon games, if you want to teach him, give him the TM flamethrower (TM35). However, if you waited till lv 50, a growlithe can indeed learn flamethrower on its own. Unfortunately, arcanine would not learn extremespeed as a result. Ah but it does, if you get it up to level 50 it will learn flamethrower then evolve it using a fire stone and when it has evolved it will learn extremespeed soz if answer is a bit late
it is very rare that a furret can even learn extremespeed.
Unfortunately, Arcanine does not learn Flamethrower. Your must level your Growlithe up to lv 50 to learn Flamethrower and then evolve it to Arcanine in order for Arcanine to have the move.
at level 51 it will learn extremespeed