The system of linear equations
$$3x - 2y - kz = 10$$
$$2x - 4y - 2z = 6$$
$$x + 2y - z = 5m$$
is inconsistent if:

The system of equations is $$3x - 2y - kz = 10$$, $$2x - 4y - 2z = 6$$, and $$x + 2y - z = 5m$$.

The determinant of the coefficient matrix is $$\begin{vmatrix} 3 & -2 & -k \\ 2 & -4 & -2 \\ 1 & 2 & -1 \end{vmatrix}$$.

Expanding along the first row: $$3((-4)(-1) - (-2)(2)) - (-2)((2)(-1) - (-2)(1)) + (-k)((2)(2) - (-4)(1))$$.

This gives $$3(4 + 4) + 2(-2 + 2) - k(4 + 4) = 24 + 0 - 8k = 24 - 8k$$.

For the system to be inconsistent, the determinant must be zero, so $$24 - 8k = 0$$, giving $$k = 3$$.

With $$k = 3$$, the equations become $$3x - 2y - 3z = 10$$, $$2x - 4y - 2z = 6$$, and $$x + 2y - z = 5m$$.

Performing $$R_1 - 3R_3$$: $$-8y = 10 - 15m$$. Performing $$R_2 - 2R_3$$: $$-8y = 6 - 10m$$.

For inconsistency, these must give different values: $$10 - 15m \neq 6 - 10m$$, which gives $$4 \neq 5m$$, so $$m \neq \frac{4}{5}$$.

Hence, the correct answer is Option A.