We consider an inverse problem for identifying a time-dependent thermal diffusivity alpha(t) in a heat conduction equation T-r(x, t) = alpha(t)T-xx(x,t), with the aid of an extra measurement of temperature at an internal point of a rod. Because the data are acquired at an inner point we require to develop a two-stage Lie-group shooting method (TSLGSM) to solve this inverse problem. The present approach is novel and is examined through some numerical tests. By comparing the calculated results with exact ones we can assure that the TSLGSM is an accurate and efficient method, whose estimation error is small even for the identification of a discontinuous and oscillatory thermal diffusivity. Under noise, the identified solutions are also acceptable. (C) 2010 Elsevier Ltd. All rights reserved.