Suppose I want to throw an axe, so first I spin it vertically by holding the end of the axe's handle with my extended arm, and then I let go of the axe when it is at it's highest point (when the handle is perpendicularly facing the ground). Because its tangential velocity is at a 90 degrees angle, the axe will fly forwards away from me, and it will spin around its center of mass while it's in the air.
My question is: will the angular velocity of the rotation of the axe after I let it go be the same as the angular velocity that I spin it with? Am I correct in saying: while I spin the axe and hold the end of the handle, the axe's moment of inertia around this axis will be bigger than the axe's moment of inertia as it is rotating around its center of mass while it's flying. According to the conservation of angular momentum I1*ω1 = I2*ω2, and because I1 > I2, ω1 will be smaller than ω2. Therefore, when I let go of the axe, it's angular velocity will increase (the time it will take the axe to rotate around its center of mass will be shorter than the time it took me to give it one full rotation while holding it by the end of the handle). Is this correct?