Basement AC problem outside blower stops when #2 Comp online

Sequence should be, outside blower and compressor #1 start. Inside blower comes a few seconds later. 15-20 seconds later outside blower shuts off and #2 compressor starts. A second or 2 later outside blower comes on again.

Yours is doing everything correctly except to restart the outside blower. Could be a control board, there are lights on the control board that will tell you what it 'thinks' it's doing or if there is a problem. A google search should get you the service manual with troubleshooting steps.