Group delay using cross correlation? An input signal + (thermal) noise travels through a channel and can be measured at its output.

Is it possible to use cross correlation between input and output signal to measure group delay? If not, is there a better way to do this?
You can use the cross correlation to measure the time delay between a signal and a delayed version of itself.

You can use cross correlation to estimate  the transfer function.

You probably would benefit the most from PHAT

https://www.mathworks.com/matlabcentral/fileexchange/26256-calculate-ambiguity-function-and-utilities?focused=5143816&amp;tab=function