Sebastian Holmes is an agent with British Intelligence.
That purposely vague description for his employer definitely fits the situation here because at the time that Holmes first becomes involved with it, it does not really exist. It might be better to state that he goes to work for the mysterious Mycroft Holmes and the group that answers to his directions.
As people who have read about that latter individual from the chronicles of Dr. John Watson are aware, Mycroft Holmes is a "government official" who, as Watson's good friend, Sherlock Holmes, describes, "is under the British government. You would also be right in a sense if you said that occasionally he is the British government".
Other chroniclers have gone to considerable length to show how Mycroft's actions were the precursors to the later establishments of MI-5 and MI-6 and the person doing the telling of Sebastian Holmes is definitely in agreement. To paraphrase the words of his brother above, in the case of the actions for which Sebastian will be tasked, Mycroft is British Intelligence.
Sebastian Holmes is the youngish nephew of Mycroft. He is the acknowledged son of the legendary consulting detective, Sherlock Holmes. The latter's well known aloofness as well as his worry about his son being the target for his enemies caused him to leave the boy in the care of his brother who was listed officially as his father and who saw to the upbringing and the proper schooling of Sebastian.
At the time of the first recorded adventure, Sebastian is a recent graduate of the University in Heidelberg having gone there after completing his proscribed studies at the Sorbonne and prior to that his attendance at King's College in Cambridge.
Returning to London now that he is ready to make something of himself, Sebastian Holmes is very much in the mood to show his illustrious father, as well as his much adored Uncle, that he is every bit as good as they were at his age, if not even at the age they are currently. This is not to say that he truly is - just that he would like to be thought as such by them.
The younger Holmes is the child of Sherlock Holmes and a woman named Matilde Huber, described as a free spirited, feisty young woman who was working as a nurse in a facility near the infamous Reichenbach Falls in Switzerland. It was she who was the primary caregiver when a very injured and near death Sherlock was brought to the hospital after his battle with Professor Moriarty. It was during the very long recovery period that Sherlock had to endure that they began their relationship which resulted in the conception of the infant. Much to Sherlock's dismay, Matilde succumbed to blood loss during delivery and a distraught Sherlock was left with a child he was in no condition to manage. The ever resourceful Mycroft with his army of assistants stepped in.
Sebastian Holmes will, in the two adventures we have of him, prove that while he is very young and often naive, is truly a smart and capable agent. He makes mistakes but he covers for them well and more importantly, he learns from them.