TTP has issued statements in the past that LET is attacking them in Kunar and is using name of Afghan Taliban. LET has indeed presence in Kunar province. there were also reports of their fighters getting killed in ISAF airstrikes in Kunar. Also LET itself was formed in Kunar province in 1990 (with base in Mureekay near Lahore), so they have some operational base in some villages of kunar, with support from local Wahabi/ahle-hadith elements.
Lashkar-e-Taiba
Kunar and Nuristan are not controlled by TTP , the two provinces are under shadow governors of Afghan Taliban, they are the most numerous and powerful ones. Afghan Taliban are the ones who have allowed TTP as well as LET in the region, even though both groups are enemies of each other but they are supporting Afghan taliban in their war against Afghan and NATO forces.