Let $\vec{A} = 2\hat{i}+4\hat{j}-2\hat{k}, \qquad \vec{B} = \hat{i}+2\hat{j}+\alpha \hat{k}.$

The projection of $\vec{A}$ on $\vec{B}$ is zero.

Scalar projection is proportional to the dot product: $\text{projection of }\vec{A}\text{ on }\vec{B} = \frac{\vec{A}\cdot \vec{B}}{|\vec{B}|}.$

For this to be zero, we must have $\vec{A}\cdot \vec{B} = 0.$

Compute the dot product: $\vec{A}\cdot \vec{B} = (2)(1) + (4)(2) + (-2)(\alpha).$

$= 2 + 8 - 2\alpha = 10 - 2\alpha.$

Set it equal to zero: $10 - 2\alpha = 0$ $2\alpha = 10$ $\alpha = 5.$

Therefore, the required integer value is $\boxed{5}.$