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We show that a continuous one-parameter group {at} of automorphisms of a separableC*-algebraA satisfies a spectrum condition if and only if it is the limit, pointwise onA, uniformly on compact subsets of IR of a sequence of inner automorphism groups whose positive generators do not increase too fast. Moreover we prove in this case that if p is a surjective morphism of a separableC*-algebraB onA then there is a similar group<img src="/fulltext-image.asp?format=htmlnonpaginated&src=U48K5715N8642032_html\220_2005_Article_BF01609053_TeX2GIFIE1.gif" border="0" alt="$$\{ \bar \alpha _t \} $$" /> of automorphisms ofB such that<img src="/fulltext-image.asp?format=htmlnonpaginated&src=U48K5715N8642032_html\220_2005_Article_BF01609053_TeX2GIFIE2.gif" border="0" alt="$$\pi  \circ \bar \alpha _t  = \alpha _t  \circ \pi $$" /> for allt.

Groups of automorphisms with spectrum condition and the lifting problem