If the average weight of $6$ students is $50\text{kg}$; that of $2$ students is $53\text{kg}$; and that of $2$ students is $55\text{kg}$; then the average weight of all students is:

Total number of students $=6+2+2=10$.
Now, calculate the total sum of the weights of all students:

$$\text{Total Weight}=(6\times 50)+(2\times 53)+(2\times 55)$$

$$\text{Total Weight}=300+106+110=516\text{kg}$$

The overall average weight is given by dividing the total weight by the total number of students:

$$\text{Average Weight}=\frac{516}{10}=51.6\text{kg}$$