The accelerated failure time model is frequently used in survival analysis because of its direct physical interpretation. Semiparametric inference methods have been extensively investigated for this model. However, the accelerated failure time model and the existing inference methods assume homogeneity of the survival data after taking log-transformation. This assumption is not always appropriate because heterogeneous data are often encountered in practice. In dealing with this heterogeneity, Yu, Yu, and Liu proposed a parametric quasi-likelihood method by assuming a known variance function, which may not be realistic for real data. In this paper, we extend the parametric quasi-likelihood method to semiparametric via relaxing its assumption and approximating the unknown variance function by using fractional polynomials approach. Simulations show that this novel extension performs superior to other methods in statistical properties of unbiasedness, efficiency, and correct coverage probability in finite samples. An application to real data set in primary biliary cirrhosis demonstrates the applicability of this new methodology. Copyright © 2012 John Wiley & Sons, Ltd.