Given \(\overrightarrow{OA} = 2\mathbf{i} - 5\mathbf{j} - 2\mathbf{k}\) and \(\overrightarrow{OB} = 4\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}\).

First, find \(\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = (4 - 2)\mathbf{i} + (-4 + 5)\mathbf{j} + (2 + 2)\mathbf{k} = 2\mathbf{i} + \mathbf{j} + 4\mathbf{k}\).

Since \(\overrightarrow{AB} = \overrightarrow{BC}\), we have \(\overrightarrow{AC} = 2\overrightarrow{AB} = 4\mathbf{i} + 2\mathbf{j} + 8\mathbf{k}\).

Thus, \(\overrightarrow{OC} = \overrightarrow{OA} + \overrightarrow{AC} = (2\mathbf{i} - 5\mathbf{j} - 2\mathbf{k}) + (4\mathbf{i} + 2\mathbf{j} + 8\mathbf{k}) = 6\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}\).

The magnitude of \(\overrightarrow{OC}\) is \(\sqrt{6^2 + (-3)^2 + 6^2} = \sqrt{36 + 9 + 36} = \sqrt{81} = 9\).

Therefore, the unit vector in the direction of \(\overrightarrow{OC}\) is \(\frac{1}{9}(6\mathbf{i} - 3\mathbf{j} + 6\mathbf{k})\).