Use grout cutting tools to remove the wrong color grout from between the tiles. A rotary tool designed for removed grout, with the appropriate sized bit, works very well (Image 1).
Finish removing the grout with a carbide grout cutter (Image 2).
Clean out the grout lines with a vacuum cleaner.
Use a tile grout sample kit to match to the original grout color.
Mix up a small amount of grout and let it sit for 10 minutes to slack.
Apply the grout into the cleaned grout lines, using a rubber grout float (Image 1).
Wipe away the excess grout with the edge of the float, holding the tool at a 45-degree angle (Image 2).
Let the grout just begin to harden then, before it hardens fully, wipe off the grout from the surface of the tile with a damp sponge (Image 1 and 2). A sponge with larger pores is most effective for this. Wipe lightly so as not to remove the grout from between the tiles.
Once the grout is mostly hard, wipe off the surface again with the sponge.
After the grout hardens, wipe off the haze left on the tile surface with a clean soft cloth.
Avoid walking on the newly grouted area several hours.