During the time of which Mughal Emperor did the English East India Company establish its first factory in India?

The first permanent English factory in India was established at Surat during the reign of Emperor Jahangir. After Captain Thomas Best’s naval victory over the Portuguese in 1612, Jahangir granted the English permission in early 1613 to set up a factory at Surat (under Thomas Aldworth), marking the Company’s first settled trading post on the subcontinent [1]. Secondary summaries of Company settlements likewise record the foundation of the Surat factory in 1613 (noting the Company’s formation in 1600 but its first Indian factory coming in the 1610s), and place Sir Thomas Roe’s later embassy to Jahangir (1615–19) in the same period of English establishment and expansion in Mughal domains [2].

Sources

1. [1] Rajiv Ahir. A Brief History of Modern India (2019 ed.). SPECTRUM. > Chapter 3: Advent of the Europeans in India > Foothold in West and South > p. 38
2. [2] Rajiv Ahir. A Brief History of Modern India (2019 ed.). SPECTRUM. > Chapter 3: Advent of the Europeans in India > Why the English Succeeded against Other European Powers > p. 57