Total distance and total time
Uphill: $6$ miles, $2$ miles/hour takes $3$ hours

Downhill: $6$ miles, $4$ miles/hour takes $1\frac12$ hours

Average speed: $\dfrac{6+6\text{ miles}}{3 + 1\frac12\text{ hours}} = \dfrac{12}{4\frac12}$ miles per hour
                                                     $=\frac{24}{9}=\frac83=2\frac23$ miles per hour


Weighted average
Uphill: 2 miles per hour
Downhill: 4 miles per hour
Takes twice as long to go up as down
$\Rightarrow$ spends twice as long going up as down
$\Rightarrow$ spends twice as long travelling at 2 miles per hour as at 4 miles per hour

$\therefore$ average speed $=\frac{2+2+4}{3}=\frac83 = 2\frac23$ miles per hour