Ezekiel 36:20
And when they entered unto the heathen, whither they went, they profaned my holy name, when they said to them, These are the people of the LORD, and are gone forth out of his land. And when they entered unto the heathen, whither they went, they profaned my holy name, when they said to them, These are the people of the LORD, and are gone forth out of his land.
When Israel were scattered and entered in among the heathen nations where they went into exile, they profaned God's holy name there. Their profaning was not hidden but took place openly among the nations into whose lands they came.
They did so by a course of conduct that copied practices displeasing to God, living in a way that contradicted their identity as the Lord's people. To bear His name while imitating what offends Him was itself the dishonoring of that holy name.
Their public disgrace after sin and under judgment also brought dishonor to Him. The nations looked upon them and said in effect that these were the people of the Lord yet had been driven forth out of His land. That spectacle of a judged and expelled people still identified with Him made His name appear dishonored among the heathen.