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In this paper, a notion of a principal $2$-bundle over a Lie groupoid hasbeen introduced. For such principal $2$-bundles, we produced a short exactsequence of VB-groupoids, namely, the Atiyah sequence. Two notions ofconnection structures viz. strict connections and semi-strict connections on aprincipal $2$-bundle arising respectively, from a retraction of the Atiyahsequence and a retraction up to a natural isomorphism have been introduced. Weconstructed a class of principal $\mathbb{G}=[G_1\rightrightarrowsG_0]$-bundles and connections from a given principal $G_0$-bundle$E_0\rightarrow X_0$ over $[X_1\rightrightarrows X_0]$ with connection. Anexistence criterion for the connections on a principal $2$-bundle over aproper, \'etale Lie groupoid is proposed. The action of the $2$-group of gaugetransformations on the category of strict and semi-strict connections has beenstudied. Finally we noted an extended symmetry of the category of semi-strictconnections.

Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid