Is it permissible to lie to one's family to conceal a good deed, such as saying "I was with my friend" when cleaning the mosque's restrooms, in order to achieve sincerity?

The way to achieve sincerity (ikhlas) is to be keen on performing good deeds in secret. The intention is to conceal as much as possible of one's acts of obedience, not to lie. The righteous predecessors - may Allah have mercy on them - were keen on concealing their good deeds. Examples include: a man having a private time during which he prays and instructs his family not to tell anyone about it; concealing tears shed out of fear of Allah by using indirect speech; one of them memorizing the entire Quran without people knowing about it; spending large sums of money without people knowing; performing long prayers in his home without people knowing; standing in the prayer line with tears flowing without the person next to him noticing; and performing night prayers secretly.