A tune will speed up the shifting assuming there's nothing wrong with the transmission. While the car is performing a shift the ECU reduces engine power to make it smoother. If for some reason the transmission is shifting too slowly or, slower than normal, that will present as an extended lag in power. Do you have a sport mode on the 2011? If so what happens in that mode?
2013 GT Premium Brembo
MGW, Blow Fish, AM Hood Struts
Tinted, Borla Touring AB, ProCal Tune
Full Shelby/Kicker Audio Upgrade