API documentation is often the most trusted resource for programming. Many approaches have been proposed to augment API documentation by summarizing complementary information from external resources such as Stack Overflow. Existing extractive-based summarization approaches excel in producing faithful summaries that accurately represent the source content without input length restrictions. Nevertheless, they suffer from inherent readability limitations. On the other hand, our empirical study on the abstractive-based summarization method, i.e., GPT-4, reveals that GPT-4 can generate coherent and concise summaries but presents limitations in terms of informativeness and faithfulness. We introduce APIDocBooster, an extract-then-abstract framework that seamlessly fuses the advantages of both extractive (i.e., enabling faithful summaries without length limitation) and abstractive summarization (i.e., producing coherent and concise summaries). APIDocBooster consists of two stages: (1) \textbf{C}ontext-aware \textbf{S}entence \textbf{S}ection \textbf{C}lassification (CSSC) and (2) \textbf{UP}date \textbf{SUM}marization (UPSUM). CSSC classifies API-relevant information collected from multiple sources into API documentation sections. UPSUM first generates extractive summaries distinct from the original API documentation and then generates abstractive summaries guided by extractive summaries through in-context learning. To enable automatic evaluation of APIDocBooster, we construct the first dataset for API document augmentation. Our automatic evaluation results reveal that each stage in APIDocBooster outperforms its baselines by a large margin. Our human evaluation also demonstrates the superiority of APIDocBooster over GPT-4 and shows that it improves informativeness, relevance, and faithfulness by 13.89\%, 15.15\%, and 30.56\%, respectively.