Infinitely many

we can restrict ourselves to non-negative numbers.

Let's choose a triplet of non-negative integers $(x,y,z)$ satisfying this equation and with $max(x,y,z) > 0$ as small as possible.

If $x^6+2y^6=4z^6$ then $x$ must be even, $x=2x_1.$

So: $32x_1^6+y^6=2z^6$ so $y=2y_1$

which mean $16x_1^6+32y_6^6=z^6$ so $z=2z_1$

that return to the original equation $x_1^6+2y_1^6=4z_1^6$

But $max(a_1,b_1,c_1) < max(a,b,c).$

This means that the only integer triple is $(0,0,0).$

Hence the answer is $1$