Fax or email her a copy of your FFL.
Have her take that copy and the pistol (along with packing material) to the FFL for an outgoing transfer.
They will charge her a fee for the transfer plus the cost of shipping.
I have looked into two near me and they charge $40 + shipping for outgoing
but only 25 for incoming
Being an FFL they can ship USPS, FedEx ground, or UPS ground.
Unless your FFL doesn't accept shipments from non-FFLs your aunt can mail it herself to your FFL
She would need to take it to a FedEx or UPS hub and ship it overnight.
Those companies require overnight. She can't use USPS
That would run $60-75 depending on insurance coverage
When your FFL receives the gun, fill out your 4473 and pay your transfer. Hopefully it isn't more than $25-30
So you are looking at roughly $100
Since I know this might come up, shipping a pistol as machines parts is just asking for headache.
If there are any problems during transit, the carrier is going to say FU, you didn't ship overnight like our policy requires.
Good luck filing any insurance claims