If you multiply a 10 bit ADC result by two, you don't increase the (ADC conversion) resolution. If you take 24 consecutive samples and divide by 6, you'll practically get an average of six 11 bit samples multiplied by two (or 4 averaged 10 bit results added together, which increases the resolution by 1 bit).
Take a look at this:
https://www.electro-tech-online.com/custompdfs/2013/01/doc8003-1.pdf
If you really want an average of six 12 bit samples (using 10 bit ADC), you'll need to take 96 samples, add them, and then divide by 24.