The aggregation of multiple rankings into a consensus ranking is a crucial task in various domains such as search engine results or user-based ratings. This task poses significant challenges due to its inherent complexity. The complexity of the problem stems not only from the need for exactness and efficiency, but also from the diversity of real-world scenarios, which often involve incomplete rankings and ties. Most existing methods propose a specific way to aggregate rankings. However, these methods often do not take into account different real use-case scenarios which can impact the relevance of their final result, as the congruence between the aggregated output and the expected outcome inherently depends on the context. To address the issue of context-dependency in ranking aggregation, we introduce a unifying framework that subsumes a variety of generalizations of the Kemeny score for incomplete rankings with ties and enables the design of new ones if a specific context requires it. Our framework is parameterized, allowing for different behaviors depending on the specific use case. We provide a broader scope of application to the methods encompassed by our approach, augmenting them with a larger theoretical and algorithmic structure. We establish an axiomatic study to better understand each method within our framework and present an algorithmic approach that includes exact methods, partitioning algorithms, and heuristics. Finally, we demonstrate the practical relevance of our approach through an empirical study on both real and synthetic datasets. Notably, the synthetic datasets are generated based on devised real-world scenarios, highlighting the context-dependent applicability of different Kemeny-based rank aggregation methods within our framework.